Monotone valuations on the space of convex functions

We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. Among these valuations we prove integral representation formulas for those which are, additionally, simple or homogeneous.


Introduction
The aim of this paper is to begin an exploration of the valuations defined on the space of convex functions, having as a model the valuations of convex bodies.
The theory of valuations is currently a significant part of convex geometry. We recall that, if K n denotes the set of convex bodies (compact and convex sets) in R n , a (realvalued) valuation is an application σ : K n → R that verifies the following (restricted) additivity condition σ(K ∪ L) + σ(K ∩ L) = σ(K) + σ(L) ∀ K, L ∈ K n such that K ∪ L ∈ K n , (1.1) together with σ(∅) = 0. (1.2) In the realm of convex geometry the most familiar examples of valuations are the so-called intrinsic volumes V k , k ∈ {0, 1, . . . , n}, which have many additional properties such as: invariance under rigid motions, continuity with respect to the Hausdorff metric, homogeneity and monotonicity. Note that intrinsic volumes include the volume (here denoted by V n ) itself, i.e. the Lebesgue measure, which is clearly a valuation.
A celebrated result by Hadwiger (see [6], [7], [8], [16]) provides a characterization of an important class of valuations on K n . Theorem 1.1 (Hadwiger). Every rigid motion invariant valuation on K n , which is continuous (with respect to the Hausdorff metric) or monotone, can be written as the linear combination of intrinsic volumes.
A special case of this theorem is known as the volume theorem.
Corollary 1.2. Let σ be a rigid motion invariant and continuous (or monotone) valuation on K n , which is simple, i.e. σ(K) = 0 for every K such that dim(K) < n. Then σ is a multiple of the volume: there exists a constant c ∈ R such that σ(K) = cV n (K) ∀ K ∈ K n .
These deep results gave a strong impulse to the development of the theory which, in the last decades, was enriched by a wide variety of new results and counts now a considerable number of prolific ramifications. A survey on the state of the art of this subject is presented in the monograph [16] by Schneider (see chapter 6), along with a detailed list of references.
Recently the study of valuations was extended from spaces of sets, like K n , to spaces of functions. The condition (1.1) is adapted to this situation replacing union and intersection by "max" and "min". In other words, if X is a space of functions, an application µ : for every u, v ∈ X such that u ∧ v ∈ X and u ∨ v ∈ X . Here u ∨ v and u ∧ v denote the point-wise maximum and minimum of u and v, respectively. To motivate this definition one can observe that the epigraphs of u ∨ v and u ∧ v are the intersection and the union of the epigraphs of u and v, respectively. The same property is shared by sub-level sets: for every t ∈ R we have This fact will be particularly important in the present paper.
Valuations defined on the Lebesgue spaces L p (R n ), p ≥ 1, were studied by Tsang in [17]. In relation to some of the results presented in our paper it is interesting to mention that one of the results of Tsang asserts that any translation invariant and continuous valuation µ on L p (R n ) can be written in the form where f is a continuous function subject to a suitable growth condition at infinity. The results of Tsang have been extended to Orlicz spaces by Kone, in [9]. Valuations of different types (taking values in K n or in spaces of matrices, instead of R), defined on Lebesgue, Sobolev and BV spaces, have been considered in [18], [10], [12], [13], [19], [20] and [14] (see also [11] for a survey).
Wright, in his PhD Thesis [21] and subsequently in collaboration with Baryshnikov and Ghrist in [3], studied a rather different class, formed by the so-called definable functions. We cannot give here the details of the construction of these functions, but we mention that the main result of these works is a characterization of valuation as suitable integrals of intrinsic volumes of level sets. These type of integrals will have a crucial role in our paper, too.
It includes the so-called indicatrix functions of convex bodies, i.e. functions of the form where K is a convex body. Note that the function ∞, identically equal to ∞, belongs to C n . This element will play in some sense the role of the empty set. If u ∈ C n we will denote by dom(u) the set where u is finite; if u = ∞, this is a non-empty convex set, and then its dimension dim(dom(u)) is well defined and it is an integer between 1 and n. We say that a functional µ : C n → R is a valuation if it verifies (1.3) for every u, v ∈ C n such that u ∧ v ∈ C n (note that C n is closed under "∨"), and µ(∞) = 0. We are interested in valuations which are rigid motion invariant (i.e. µ(u) = µ(u • T ) for every u ∈ C n and every rigid motion T of R n ), and monotone decreasing (i.e. µ(u) ≤ µ(v) for every u, v ∈ R n such that u ≥ v point-wise in R n ). As µ(∞) = 0, we immediately have that they are non-negative in C n . We will also need to introduce a notion of continuity of valuations. In this regard, note that for (rigid motion invariant) valuations on the space of convex bodies, continuity and monotonicity are conditions very close to each other. In particular monotonicity implies continuity, as Hadwiger's theorem shows. The situation on C n is rather different and it is easy to provide examples of monotone valuations which are not continuous with respect to any reasonable notion of convergence on C n . We say that a valuation on C n is monotone-continuous (m-continuous for short) if lim i→∞ µ(u i ) = µ(u) whenever u i , i ∈ N, is a decreasing sequence in C n converging to u ∈ C n point-wise in the relative interior of dom(u), and such that u i ≥ u in R n for every i.
How does a "typical" valuation of this kind look like? A first answer is provided by functionals of type (1.3); indeed we will see in section 6 that if f : R → R is a non-negative decreasing function which verifies the integrability condition is a rigid motion invariant, monotone decreasing valuation on C n , which is moreover mcontinuous if f is right-continuous. A valuation of this form vanishes obviously on every function u ∈ C n such that dim(dom(u)) < n: dim(dom(u)) < n ⇒ µ(u) = 0. (1.7) When µ has this property we will say that it is simple. Our first characterization result is the following theorem, proven in section 8. The proof of this fact is based on a rather simple idea, even if there are several technical points to transform it into a rigorous argument. First, µ determines the function f as follows: for t ∈ R let σ t : K n → R be defined as It is straightforward to check that this is a rigid motion and monotone increasing valuation, so that by the volume theorem there exists a constant, which will depend on t and which we call f (t), such that As µ is decreasing, f is decreasing too. Now for every u ∈ C n and K ⊆ dom(u), by monotonicity we obtain This chain of inequalities, and the fact that µ is simple, permit to compare easily the value of µ(u) with upper and lower Riemann sums of f • u, over suitable partitions of subsets of dom(u). This leads to the proof of (1.6). Monotonicity is an essential ingredient of this argument. It would be very interesting to obtain a similar characterization of simple valuations without this assumption.
If we apply the layer cake (or Cavalieri) principle, we obtain a second way of writing the valuation µ defined by (1.6): where V n is the n-dimensional volume, "cl" is the closure, and ν is a Radon measure on R identified by the equality Note that the set cl({u < t}) is a compact convex set, i.e. a convex body. Formula (1.8) suggests to consider the more general expression where, for k ∈ {0, . . . , n}, V k is the k-th intrinsic volume and ν is a Radon measure on R.
We will see (still in section 6) that the integral in (1.9) is finite for every u ∈ C n if and only if (which is equivalent to (1.5) when k = n) and in this case it defines a rigid motion invariant, decreasing and m-continuous valuation on C n . Valuations of type (1.9) are homogeneous of order k in the following sense. For u ∈ C n and λ > 0, let u λ : (note that u λ ∈ C n ). Then µ(u λ ) = λ k µ(u).
In section 9 we will prove the following fact.
Theorem 1.4. Let µ be a valuation on C n wich is rigid motion invariant, decreasing, m-continuous and k-homogeneous for some k. Then necessarily k ∈ {0, 1, . . . , n} and there exists a Radon measure ν on R, verifying (1.10), such that µ can be written in the form (1.9). Theorems 1.3 and 1.4 may suggest that valuations of type (1.9) could form a sort of generators for invariant, monotone and m-continuous valuations on C n , playing a similar role to intrinsic volumes for convex bodies (with the difference that the dimension of the space of valuations on C n is infinite). On the other hand in the conclusive section of the paper we show the existence of valuations on C n with the above properties, which cannot be decomposed as the sum of homogeneous valuations, and hence are not the sum of valuations of the form (1.9).
The second author would like to thank Monika Ludwig for encouraging his interest towards valuations on convex functions and for the numerous and precious conversations that he had with her on this subject.

Preliminaries
We work in the n-dimensional Euclidean space R n , n ≥ 1, endowed with the usual Euclidean norm | · | and scalar product (·, ·). For x 0 ∈ R n and r > 0, B r (x 0 ) denotes the closed ball centred at x 0 with radius r; when x 0 = 0 we simply write B r . For k ∈ [0, n], the k-dimensional Hausdorff measure is denoted by H k . In particular H n denotes the Lebesgue measure in R n (which, as we said, will be often indicated by V n , especially when referred to convex bodies). Integration with respect to such measure will be always denoted simply by dx, where x is the integration variable. Given a subset A of R n we denote by int(A) and cl(A) its interior and its closure, respectively.
As usual, we will denote by O(n) and SO(n) respectively, the group of rotations and of proper rotations of R n . By a rigid motion we mean the composition of a rotation and a translation, i.e. a mapping T : R n → R n such that there exist R ∈ O(n) and x 0 ∈ R n for which

Convex bodies
A convex body is a compact convex subset of R n . We will denote by K n the family of convex bodies in R n . For all the notions and results concerning convex bodies we refer to the monograph [16]. The set K n can be endowed with a metric, induced by the Hausdorff distance (see [16] for the definition). Let K ∈ K n ; if int(K) = ∅, then K is contained in some k-dimensional affine sub-space of R n , with k < n; the smallest k for which this is possible is called the dimension of K, and is denoted by dim(K). Clearly, if K has non-empty interior we set dim(K) = n. Using this notion we can define the relative interior of K as the subset of those points x of K for which there exists a k-dimensional ball centred at x and contained in K, where k = dim(K). The relative interior will be denoted by relint(K). The notion of relative interior can be given in the same way for every convex subset of R n . K n can be naturally equipped with an addition (Minkowski, or vector, addition) and a multiplication by non-negative reals. Given K, L ∈ K n and s ≥ 0 we set K n is closed with respect to these operations.
To every convex body K ∈ K n we may assign a sequence of (n + 1) numbers, V k (K), k = 0, . . . , n, called the intrinsic volumes of K; for their definition see [16,Chapter 4]. We recall in particular that V n (K) is the volume, i.e. the Lebesgue measure, of K, while V 0 (K) = 1 for every K ∈ K n \ {∅}. More generally, if K is a convex body in R n having dimension k ∈ {0, 1, . . . , n}, then V k (K) is the k-dimensional Lebesgue measure of K as a subset of R k . As real-valued functionals defined on K n , intrinsic volumes are continuous, monotone increasing with respect to set inclusion and invariant under the action of rigid motions: V i (T (K)) = V i (K) for every i ∈ {0, . . . , n}, K ∈ K n and for every rigid motion T . Moreover, the intrinsic volumes are special and important examples of valuations on the space of convex bodies. We recall that a (real-valued) valuation on K n is a mapping σ : K n → R such that σ(∅) = 0 and The following characterization theorem of Hadwiger (see, for instance, [16, Chapter 6]) will be a crucial tool in this paper.
Theorem 2.1. Let σ be a valuation on K n which is invariant with respect to rigid motions, and either continuous with respect to the Hausdorff metric or monotone with respect to set inclusion. Then σ is the linear combination of intrinsic volumes, i.e. there exist c 0 , . . . , c n ∈ R, such that Moreover, if σ is increasing (resp. decreasing) then c i ≥ 0 (resp. c i ≤ 0) for every i ∈ {0, . . . , n}.
3 The space C n Let us a consider a function u : R n → R ∪ {∞}, which is convex. We denote the so-called domain of u as dom(u) = {x ∈ R n : u(x) < ∞}.
By the convexity of u, dom(u) is a convex set. By standard properties of convex functions, u is continuous in the interior of dom(u) and it is Lipschitz continuous in any compact subset of int(dom(u)). We will sometimes use the following notation, for u ∈ C n , Ω u = int(dom(u)).
In this work we focus in particular on the following space of convex functions: Here by l.s.c. we mean lower semi-continuous, i.e.
lim inf Note that the function ∞ (which, we recall, is identically equal to ∞ on R n ) belongs to our functions space. As it will be clear in the sequel, this special function plays the role that the empty set has for valuations defined on families of sets (instead of functions).
Remark 3.1. Let u ∈ C n . As a consequence of convexity and the behavior at infinity we have that inf R n u > −∞. Moreover, by the lower semi-continuity, u admits a minimum in R n . We will often use the notation m(u) = min R n u . We will also need to consider the following subset of C n : Let A ⊆ R n ; we denote by I A : R n → R ∪ {∞} the so-called indicatrix function of A, which is defined by If K ⊂ R n is a convex body, then I K ∈ C n .
Sub-level sets of functions belonging to C n will be of fundamental importance in this paper. Given u ∈ C n and t ∈ R we set and Both sets are empty for t < m(u). K t is a convex body for all t ∈ R, by the properties of u. For all real t, Ω t is a bounded (possibly empty) convex set, so that its closure cl(Ω t ) is a convex body, obviously contained in K t .
Lemma 3.2. Let u ∈ C n ; for every t > m(u) Proof. We start by considering the case in which dim(K t ) = n. Assume by contradiction that there exists a point x ∈ int(K t ) such that u(x) = t. Then x is a local maximum for u but, by convexity, this is possible only if u ≡ t in K t , which, in turn implies that t = m(u), a contradiction. If dim(K t ) = k < n then, by convexity, dom(u) is contained in a k-dimensional affine subspace H of R n , and we can apply the previous argument to u restricted to H to deduce the assert of the lemma. Corollary 3.3. Let u ∈ C n ; for every t > m(u) cl(Ω t ) = K t .

On the intrinsic volumes of sub-level sets
As we have just seen, if u ∈ C n and t ∈ R, the set Ω t = {u < t} is empty for t ≤ m(u) and it is a bounded convex set for t > m(u). For k ∈ {0, . . . , n}, we define the function v k (u; ·) : R → R as follows v k (u; t) = V k (cl(Ω t )).
As intrinsic volumes are non-negative and monotone with respect to set inclusion and the set Ω t is increasing with respect to inclusion as t increases, v k (u; ·) is a non-negative increasing function. In particular it is a function of bounded variation, so that there exists a (non-negative) Radon measure on R, that we will denote by β k (u; ·), which represents the weak, or distributional, derivative of v k (see for instance [2]).
We want to describe in a more detailed way the structure of the measure β k . In general, the measure representing the weak derivative of a non-decreasing function consists of three parts: a jump part, a Cantor like part and an absolutely continuous part (with respect to Lebesgue measure). We will see that β k does not have a Cantor part and its jump part, if any, is a single Dirac delta at m(u).
As a starting point, note that as v i is identically zero in (−∞, m(u)] then β k (u; η) = 0 for every measurable set η ⊆ (−∞, m(u)). On the other hand, in (m(u), ∞), due to the Brunn-Minkowski inequality for intrinsic volumes, the function v k have a more regular behavior than that of a non-decreasing function. Indeed, for k ≥ 1, let t 0 , t 1 ∈ (m(u), ∞) and consider, for λ ∈ [0, 1], t λ = (1 − λ)t 0 + λt 1 . Then we have the set inclusion which follows from the convexity of u. By the monotonicity of intrinsic volumes and the Brunn-Minkowski inequality for such functionals (see [16,Chapter 7]), and by Corollary In other words, the function v k to the power 1/k is concave in (m(u), ∞). This implies in particular that v k is absolutely continuous in (m(u), ∞) so that the measure β k is absolutely continuous with respect to the Lebesgue measure in this interval, and its density is given by the point-wise derivative of v k , which exists a.e. (see [2,Chapter 3]). Next we examine the behavior at m(u); as v k is constantly zero in (−∞, m(u)] . On the other hand let t i , i ∈ N, be a decreasing sequence converging to m(u), with t i > m(u) for every i, and consider the corresponding sequence of convex bodies L i = cl(Ω t i ), i ∈ N. This is a decreasing sequence and L i ⊇ K m(u) for every i. Moreover, trivially This implies in particular that K m(u) is the limit of the sequence L i with respect to the Hausdorff metric (see [16,Section 1.8]). Then then v k (u; t) has a jump discontinuity at m(u) of amplitude V k (K m(u ). In other words The case k = 0 can be treated as follows: as V 0 (K) is the Euler characteristic of K for every K, i.e. is constantly 1 on K n \ {∅}, v 0 (u; t) equals 0 for t ≤ m(u) and equals 1 for t > m(u); hence β 0 is just the Dirac point mass measure concentrated at m(u).
The following statement collects the facts that we have proven so far in this part.
Proposition 3.4. Let u ∈ C n and k ∈ {0, . . . , n}; let K t , v k be defined as before. Define the measure β k as where δ denote the Dirac point-mass measure (and H 1 is the Lebesgue measure on R). Then β k (u; ·) is the distributional derivative of v k , more precisely v k (u; t) = β k (u; (m(u), t]) ∀ t ≥ m(u) and v k (u; t) = 0 ∀t ≤ m(u).
In particular v k (u; ·) is left-continuous at m(u).

Max and min operations in C n
As we will see, the definition of valuations on C n is based on the point-wise minimum and maximum of convex functions. This part is devoted to some basic properties of these operations. Given u and v in C n we set, for x ∈ R n , Hence u ∨ v and u ∧ v are functions defined in R n , with values in R ∪ {∞}.
Remark 3.5. If u, v ∈ C n then u∨v belongs to C n as well. Indeed convexity and behavior at infinity are straightforward. Concerning lower semicontinuity of u∨v, this is equivalent to say that {u ∨ v ≤ t} is closed for every t ∈ R, which follows immediately from the equality On the contrary, u, v ∈ C n does not imply, in general, that u ∧ v ∈ C n (a counterexample is given by the indicatrix functions of two disjoint convex bodies).
For u, v ∈ C n and t ∈ R the following relations are straightforward: In the sequel we will also need the following result.
Proposition 3.6. Let u, v ∈ C n be such that u ∧ v ∈ C n . Then, for every t ∈ R, Proof. Equality (3.15) comes directly from the second equality in (3.13), passing to the closures of the involved sets. As for the proof of (3.14), we first observe that (see the next lemma). Let t > m(u ∨ v); then, by Corollary 3.3 and (3.12): If we assume that t ≤ m(u ∨ v), then we have t ≤ m(u) or t ≤ m(v). In the first case {u < t} = ∅, so that the left hand-side of (3.14) is empty. On the other hand u ∨ v ≤ u implies that the right hand-side is empty as well. The case t ≤ m(v) is completely analogous.
Lemma 3.7. If u, v ∈ C n are such that u ∧ v ∈ C n , then where the last relation comes from the assumption u ∧ v ∈ C n . Hence {u ≤ t} and {v ≤ t} are non-empty convex bodies such that their union is also a convex body. This implies that they must have a non-empty intersection. But then We conclude this section with a proposition (see [4, Lemma 2.5]) which will be frequently used throughout the paper.
Proposition 3.8. If u ∈ C n there exist two real numbers a and b, with a > 0, such that 4 Valuations on C n Definition 4.1. A valuation on C n is a map µ : C n → R such that µ(∞) = 0 and A valuation µ is said: • rigid motion invariant, if µ(u) = µ(u • T ) for every u ∈ C n and for every rigid motion T ; • monotone decreasing (or just monotone), if µ(u) ≤ µ(v) whenever u, v ∈ C n and u ≥ v point-wise in R n ; • k-simple (k ∈ {1, . . . , n}) if µ(u) = 0 for every u ∈ C n such that dim(dom(u)) < k; • simple, if µ is n-simple, i.e. if µ(u) = 0 for every u such that dom(u) has no interior points.
The following simple observation will turn out to be very important.
Remark 4.2. Every monotone decreasing valuation µ on C n is non-negative. If we set ∞(x) = ∞ for all x ∈ R n , then u ≤ ∞ holds for each u ∈ C n , which in turn leads to µ(u) ≥ µ(∞) = 0 by monotonicity.
In the sequel other features of valuations will be considered, like monotone-continuity and homogeneity. Concerning the latter, the definition is the following. Definition 4.3. Let µ be valuation on C n and let α ∈ R; we say that µ is positively homogeneous of order α, or simply α-homogeneous, if for every u ∈ C n and every λ > 0 we have This corresponds to homogeneity with respect to a vertical stretching of the graph of u, while Definition 4.3 involves a horizontal stretching. In addition, one could consider a more general type of homogeneity where both types of dilations (vertical and horizontal) are simultaneously in action. Definition 4.3 is more natural from the point of view of convex bodies. Indeed, if u = I K with K ∈ K n , then u λ is the indicatrix function of the dilated body λK.
The next one is the definition of monotone-continuous valuations.
Definition 4.5. Let µ be a valuation on C n ; µ is called monotone-continuous, or simply m-continuous, if the following property is verified: given a sequence u i ∈ C n , i ∈ N, and u ∈ C n , such that: We recall that "relint" denotes the relative interior of a convex set (see the definition given in section 2.1).
Remark 4.6. Let K i , i ∈ N, be a sequence converging to a convex body K in the Hausdorff metric. Assume moreover that the sequence is monotone increasing: Then the corresponding sequence of indicatrix functions I K i , i ∈ N, is decreasing, it verifies I K i ≥ I K point-wise in R n , for every i, and Hence, if µ is an m-continuous valuation on C n , lim i→∞ µ(I K i ) = µ(I K ).

Geometric densities
Throughout this section, µ will be a rigid motion invariant and monotone decreasing valuation on C n .
Let t ∈ R be fixed, and consider the following application σ t defined on K n : It is straightforward to check that σ t is a rigid motion invariant valuation on K n . Moreover, if K ⊆ L, then I K ≥ I L , so that σ t (K) ≤ σ t (L), i.e. σ t is monotone increasing. By Theorem 2.1 there exist (n + 1) non-negative coefficients, that we will denote by f 0 , f 1 , . . . , f n , such that The numbers f k 's clearly depend on t, i.e. they are real-valued functions defined on R; we will refer to these functions as the geometric densities of µ.
We prove that the monotonicity of µ implies that these functions are monotone decreasing. Fix k ∈ {0, . . . , n} and let B k be a closed k-dimensional ball of radius 1 in R n ; note that Fix r ≥ 0; V j is positively homogeneous of order j, hence for every t ∈ R we have Hence we get On the other hand, as µ is decreasing, the function t → µ(t + I rB k ) is decreasing for every r ≥ 0; this proves that f k is decreasing.
Proposition 5.1. Let µ be a rigid motion invariant and decreasing valuation defined on C n . Then there exists (n + 1) functions f 0 , f 1 , . . . , f n , defined on R, non-negative and decreasing, such that for every convex body K ∈ K n and for every t ∈ R If in addition to the previous assumption the valuation µ is m-continuous, then all its geometric densities are right-continuous, i.e.
Indeed, for every convex body K the function is right-continuous, by the definition of m-continuity. If we chose K = {0}, as V k (K) = 0 for k ≥ 1 and V 0 (K) = 1, we have, by (5.16) This proves that f 0 is right-continuous. If we now take K to be a one-dimensional convex body, such that V 1 (K) = 1, we have that V k (K) = 0 for every k ≥ 2, hence As the left hand-side is right-continuous and f 0 is also right-continuous (by the previous step) then f 1 must have the same property. Proceeding in a similar way we obtain that each f k is right-continuous. Assume now that µ is positively homogeneous of some order α; then it is readily checked that for every t ∈ R the valuation σ t defined at the beginning of this section is positively homogeneous of the same order, i.e.
On the other hand, each σ t is a linear combination of the intrinsic volumes V k 's, and V k is positively homogenous of order k. We are led to the following conclusion. Corollary 5.3. Let µ be a rigid motion invariant and monotone decreasing valuation on C n and assume that it is α-homogeneous for some α ∈ R. Then necessarily α ∈ {0, 1, . . . , n} and f k ≡ 0 for every k = α.
We are in position to prove a characterization result for 0-homogeneous valuations which are also monotone and m-continuous. We recall that, for u ∈ C n , m(u) is the minimum of u on R n . Proposition 5.4. Let µ be a rigid motion invariant, monotone decreasing and continuous valuation on C n and assume that it is 0-homogeneous. Then, for every u ∈ C n we have Proof. We first prove the claim of this proposition under the additional assumption that dom(u) is bounded; let K be a convex body containing dom(u). Moreover, let x 0 ∈ R n be such that u(x 0 ) = m(u). Then As µ is monotone decreasing On the other hand, by Poposition 5.1 and Corollary 5.3 we have To extend the result to the general case, for u ∈ C n and i ∈ N let where B(i) denotes the closed ball of radius i centred at the origin. The sequence u i is contained in C n , is monotone decreasing and converges point-wise to u in R n . As µ is m-continuous we have, by the previous part of the proof, On the other hand, as m(u) = min R n u, by the point wise convergence we have that for i sufficiently large m(u i ) = m(u).
Another special case in which more information can be deduced on geometric densities, is when the valuation µ is simple.
Proposition 5.5. Let µ be a rigid motion invariant, monotone decreasing and simple valuation on C n . Then, for each k ∈ {0, . . . , n − 1}, the k-th geometric density f k of µ is identically zero.
Proof. Fix t ∈ R; the valuation σ t : K n → R defined by is monotone, rigid motion invariant and simple; the volume theorem (Corollary 1.2) and the definition of geometric densities yield In other words, f k (t) = 0 for every k = 0, . . . , n − 1 and t ∈ R.
The following result relates homogeneity and simplicity, and its proof makes use of geometric densities.
Proposition 5.6. Let µ : C n → R be a valuation with the following properties: • µ is rigid motion invariant; • µ is monotone decreasing; • µ is k-homogeneous; • µ is m-continuous.
Then µ is k-simple.
Proof. Let f 0 , f 1 , . . . , f n be the geometric densities of µ. As µ is k-homogeneous, f i ≡ 0 for every i = k. Let u ∈ C n be such that dim(dom(u)) < k.
As µ is monotone we have On the other hand V k (cl(dom(u i ))) = 0, as dim(dom(u i )) < k. Hence µ(u i ) = 0 for every i. To conclude, note that u i , i ∈ N, is a decreasing sequence of functions in C n , converging to u point-wise in R n . As a consequence of m-continuity of µ we have

Regularization of geometric densities
In the sequel sometimes it will be convenient to work with valuation having geometric densities with more regularity than that of a decreasing function. In this section we describe a procedure which allows to approximate (in a suitable sense) a valuation with a sequence of valuations having smooth densities. Let g : R → R be a standard mollifying kernel, i.e. g has the following properties: g ∈ C ∞ (R), g(t) ≥ 0 for every t ∈ R, the support of g is contained in [−1, 1] and R g(t)dt = 1.
For > 0 let g : R → R be defined by Then g ∈ C ∞ (R); g (t) ≥ 0 for every t ∈ R; the support of g is contained in [− , ] and R g (t)dt = 1.
Now let u ∈ C n and consider the function φ : R → R defined by By the properties of µ, this is a non-negative and decreasing function. For > 0 and t ∈ R set: Then φ is a non-negative decreasing function of class C ∞ (R). Moreover, by the properties of the kernel g, It is a straightforward exercise to verify that µ inherits most of the properties of µ: it is a valuation, rigid motion invariant, non-negative and decreasing. Moreover, if f k, , for k ∈ {0, . . . , n}, denote the geometric densities of µ , we have . . , f n are the densities of µ. Indeed, for every convex body K ∈ K n and every t ∈ R we have The core properties of the above construction can be summed up in the following proposition.

Integral valuations
In this section we introduce a class of integral valuations which will turn out to be crucial in the characterization results that we will present in the sequel. As we will see, they are similar to those introduced by Wright in [21] and subsequently studied by Baryshnikov, Ghrist and Wright in [3]. Let ν be a (non-negative) Radon measure on the real line R and fix k ∈ {0, . . . , n}. For every u ∈ C n we set where Ω t = {u < t} for every t ∈ R. As noted in sub-section 3.1, the function t → V k (cl(Ω t )) vanishes on (−∞, m(u)] and, for k = 0 its k-th root is concave in (m(u), ∞), while for k = 0 it is simply constantly 1 in (m(u), ∞); hence it is Borel measurable. Moreover it is non-negative, so that it is integrable with respect to ν. On the other hand its integral (6.17) might be ∞. We first find equivalent conditions on ν such that (6.17) is finite for every u ∈ C n . Proposition 6.1. Let ν be a non-negative Radon measure on the real line. The integral (6.17) is finite for every u ∈ C n if and only if Proof. Assume that µ(u) is finite for every u. Choosing in particular u ∈ C n defined by u(x) = |x| we have that V k (cl(Ω t )) is zero for every t ≤ 0. For t > 0, cl(Ω t ) is a ball centred at the origin with radius t, hence V k (cl(Ω t )) = c(n, k)t k with c(n, k) > 0. Therefore Vice versa, assume that condition (6.18) holds. Given u ∈ C n there exists a > 0 and b ∈ R such that u(x) ≥ a|x| + b for every x (see Proposition 3.8). Hence, for t ∈ R, t ≥ b, while Ω t is empty for t ≤ b. By the monotonicity of intrinsic volumes and the last integral is finite by (6.18).
From now on in this section the summability condition (6.18) will always be assumed (with respect to some fixed index k ∈ {0, . . . , n}). Proposition 6.2. Let k ∈ {0, 1, . . . , n} and let t be a fixed real number. Then the appli- Proof. i) The condition on ∞ is easily verified, as a matter of fact Let now u, v ∈ C n be such that u ∧ v ∈ C n . By Proposition 3.6, for every t ∈ R we have Consequently, as intrinsic volumes are valuations, we get ii) Let u ∈ C n and let T : R n → R n be a rigid motion; let moreover v := u • T ∈ C n , i.e. v(y) = u(T (y)) for every y ∈ R n . Then, for every t ∈ R, As intrinsic volumes are invariant with respect to rigid motions, we have for every t.
iii) As for monotonicity, if u, v ∈ C n and u ≤ v point-wise on R n , then for every t ∈ R, By the monotonicity of intrinsic volumes iv) Let u ∈ C n and λ > 0, and define u λ by For t ∈ R we have Then, by homogeneity of intrinsic volumes .
v) Let u ∈ C n and let u i , i ∈ N, be a sequence in C n , point-wise decreasing and converging to u in the relative interior of dom(u). We want to prove that Let j ∈ {0, 1, . . . , n} be the dimension of dom(u); for every i ∈ N, as u i ≥ u we have that dom(u i ) ⊆ dom(u), and, in particular, the dimension of the domain of u i is less than or equal to j. If k > j, then V k (cl({u < t})) = 0 for every t ∈ R and analogously V k (cl({u i < t})) = 0 for every i, so that the assert of the proposition holds true. Hence we may assume that k ≤ j and, up to restricting all involved functions to a j-dimensional affine subspace of R n containing the domain of u, we may assume without loss of generality that j = n.
As usual, we denote min R n u by m(u). If t ≤ m(u), then, for all i, {u i < t} = {u < t} = ∅ and the claim holds trivially. Let now t > m(u), then, by Corollary 3.3 As dim(dom(u)) = n and t > m(u), K t is a convex body with non-empty interior (this follows from the convexity of u). Let Clearly K i t ⊆ K t for every i. On the other hand, if x is an interior point of K t , then u(x) < t (see Lemma 3.2). Hence u i (x) < t for sufficiently large i, which leads to As a consequence, K i t converges to K t in the Hausdorff metric, and then (by the continuity of intrinsic volumes) lim as we wanted. ii) it is rigid motion invariant; iii) it is monotone; iv) it is k-homogeneous; v) it is m-continuous.
Proof. Claims i) -iv) follow easily from Proposition 6.2 by integration. The proof of the m-continuity of µ is a bit more delicate. Let u ∈ C n and let u i , i ∈ N, be a sequence in C n , point-wise decreasing and converging to u in relint(dom(u)).
As u i ≥ u we have, for every t ∈ R, for every i ∈ N. By Proposition 6.2 we know that This fact and the monotone convergence theorem imply Let µ be a valuation of the form (6.17); by Proposition 5.1 and Corollary 5.3, µ has exactly one geometric density which is not identically zero, i.e. f k ; this can be explicitly computed in terms of the measure ν. Let K ∈ K n be such that V k (K) > 0, then, for t ∈ R We observe that this is a non-increasing function and, by the basic properties of measures, it is right-continuous.

An equivalent representation formula
As in the previous part of this section, ν will be a non-negative Radon measure on R verifying the integrability condition (6.18), where k is a fixed integer in {0, 1, . . . , n}. Moreover, f : R → R is defined by ∞)). (6.20) We first consider the case k ≥ 1. Note that (6.18) is equivalent to Now let u ∈ C n and let v k (u; ·) be the function defined in section 3.
For simplicity we set h(t) = v k (u; t) for every t ∈ R; h is a monotone non-decreasing function identically vanishing on (−∞, m(u)] and h 1/k is concave in (m(u), ∞), as pointed out in section 3.1; in particular h is locally Lipschitz. This implies (see for instance [2,Chapter 3]) that the product f h is a function of bounded variation in (m(u), ∞) and its weak derivative is the measure −hν + h f H 1 (we recall that H 1 is the one-dimensional Lebesgue measure). Note also that f h is rightcontinuous, as f has this property. Hence for every t 0 , t ∈ R, with m(u) < t 0 ≤ t, If we let t 0 → m(u) + we get Indeed, as we proved in section 3.1, By Lemma 3.8, there is a constants a > 0 such that On the other hand, the integrability condition (6.21) implies lim inf The above formula is proven for k ≥ 1. The case k = 0 is straightforward, indeed so that (6.23) becomes which is true by the definition of f . The previous considerations provide the proof of the following result.
Proposition 6.4. Let k ∈ {0, . . . , n}, let ν be a non-negative Radon measure on R verifying the integrability condition (6.18), and let f be defined as in (6.20). Let µ : C n → R be the valuation defined by (6.17). Then for every u ∈ C n Valuations expressed as in the above proposition were considered in [21] and [3].
In the remaining part of this section we analyze two special cases of the integral valuation introduced so far, corresponding to the indices k = 0 and k = n, which can be written in a simpler alternative form.

The case k = 0
Let ν be a Radon measure on R; the integrability condition (6.18) for k = 0 is just which is equivalent to saying that the function f defined by (6.20) is well defined (i.e. finite) in R. Let u ∈ C n ; as we pointed out before, Then for every u ∈ C n .
6.3 The case k = n Proposition 6.5. Let ν be a Radon measure on R verifying the integrability condition ∞ 0 t n dν(t) < ∞, let f be defined as in (6.20), and let µ be the valuation: Then Proof. Let us extend f to R ∪ {∞} setting f (∞) = 0. As a direct consequence of the so-called layer cake (or Cavalieri's) principle and of the definition of f , we have that where H n denotes the Lebesgue measure in R n . On the other hand, for every t ∈ R the set {u < t} is convex and bounded, so that its boundary is negligible with respect to the Lebesgue measure. Hence H n ({u < t}) = V n (cl({u < t})) for every t.
We can change the point of view and take the function f as a starting point, instead of the measure ν. Proposition 6.6. Let f : R → R be non-negative, decreasing, and right-continuous. Define the mapping µ : C n → R as follows for every u ∈ C n . Then: i) µ is well defined (i.e. µ(u) ∈ R for every u ∈ C n ) if and only if f verifies the following integrability condition ii) µ is a valuation on C n , and it is rigid motion invariant, simple and decreasing; iii) µ is n-homogeneous; iv) µ is m-continuous.
The proof follows directly from the previous considerations and the dominated convergence theorem. A typical example in this sense is given by the application µ defined by

A decomposition result for simple valuations
We start by introducing a particular class of partitions of convex bodies that will be used in the sequel.

Convex partitions and inductive partitions
Definition 7.1. (Convex partition). Let K, K 1 , . . . , K N ∈ K n ; Definition 7.2. (Inductive partition). Let K ∈ K n . A convex partition P of K is called an inductive partition if there exists a sequence of convex bodies H 1 , . . . , H l = K such that, for all i = 1, . . . , l, one of the two following conditions holds true: Examples. It is immediate to prove that convex partitions made of one or two elements are inductive partitions. On the other hand it is easy to construct a convex partition of three elements which is not inductive. Let K be a disk in the plane, centred at the origin, and consider three rays from the origin such that the angle between any two of them is 2π/3. These rays divide K into three subsets K 1 , K 2 and K 3 which form a convex partition P of K. P is not an inductive partition.

Complete partitions
From now on in this section P will be a polytope of R n , i.e. the convex hull of finitely many points of R n . Note in particular that P ∈ K n ; moreover we will always assume that P has non-empty interior. We consider a convex partition P = {P 1 , . . . , P N } of P whose elements are all polytopes, with non-empty interior; we will refer to such partitions as polytopal partitions.
If H is a hyperplane (i.e. an affine subspace of dimension (n − 1)) of R n , we can refine P by H in the usual way. Let H + and H − be the closed half-spaces determined by H and set   Proof. The proof proceeds by induction on N . For N = 1 the assert is true as any partition consisting of one element is trivially an inductive partition. Let N ≥ 2 and assume that the claim is true for every integer up to (N − 1). Let P be a complete polytopal partition of P . Let H be a hyperplane containing a (n − 1)-dimensional facet of an element of P, intersecting the interior of P . Such a hyperplane exists because P has non-empty interior and N ≥ 2. Let H + and H − be the closed half-spaces determined by H. Then, as P is complete (and each P i has non-emtpy interior), each P i is contained either in H + or in H − . Moreover, as N ≥ 2 and H ∩ int(P ) = ∅, there exist at least one element of P contained in H + and at least one element in H − . Then clearly Each of these partitions has a number of elements which is strictly less than N . Consequently, by the induction hypothesis, P + is also an inductive partition of P ∩ H + and H − is an inductive partition of P ∩ H − . Therefore, by Definition 7.2, there exist two sequences P + 1 , . . . , P + j = P ∩ H + and P − 1 , . . . , P − k = P ∩ H − that fulfill the required properties. We claim that such a sequence can be formed for the partition P as well: as a matter of fact consider the following P + 1 , . . . , P + j , P − 1 , . . . , P − k , P.
As P + j ∪ P − k = P and int(P + j ∩ P − k ) = ∅ we conclude that P is an inductive partition too.

Rectangular partitions
A rectangle R in R n is a set of the form where, for j = 1, . . . , n, a j and b j are real numbers such that a j < b j . In particular, R is a convex polytope, and each of its facets is parallel to a hyperplane of the form e ⊥ j , for some j ∈ {1, . . . , n} (where {e 1 , . . . , e n } is the canonical basis of R n ). This property characterizes rectangles.
A rectangular partition of a rectangle R is a partition If P is a rectangular partition of a rectangle R, and we refine it so that its refinement P is complete, as indicated in Remark 7.4, then P is still a rectangular partition; indeed each facet of each element of P is contained in a hyperplane parallel to e ⊥ j , for some j.

A decomposition result for simple valuations
The following result is the main motivation for the definition of inductive partitions.
Lemma 7.6. Let µ be a simple valuation on C n . Let K be a convex body and let P = {K 1 , . . . , K N } be an inductive partition of K. Then, for every u ∈ C n , Proof. Since P is an inductive partition we can find a sequence of convex bodies H 1 , . . . , H l = K with the properties stated in Definition 7.2. We argue by induction on l. If l = 1 the claim holds trivially. Assume now that the claim is true up to l − 1. If H l ∈ P we can conclude as in the case l = 1. Therefore we may assume ∃j, k < l such that H j ∪ H k = K and int(H j ∩ H k ) = ∅. As H j and H k are convex bodies, u + I H j and u + I H k belong to C n . Moreover In particular, as int(H j ∩ H k ) is empty, dim(dom(u + I H j ∩H k )) < n. Hence, as µ is a simple valuation, we get Now, if we set P = {P ∈ P : P ⊆ H j } and P = {P ∈ P : P ⊆ H k } and apply the inductive hypothesis to the just defined partitions we get µ(u + I H j ) + µ(u + I H k ) = P ∈P µ(u + I P ) + P ∈P µ(u + I P ) = P ∈P µ(u + I P ).

Characterization results I: simple valuations
Our first characterization result is a converse of Proposition 6.6.
Theorem 8.1. Let µ : C n → R be a valuation with the following properties: i) µ is rigid motion invariant; ii) µ is monotone decreasing; iii) µ is simple; iv) µ is m-continuous.
Then there exists a function f : R → R, non-negative, decreasing, right-continuous and verifying the integrability condition where ν is the Radon measure related to f by: The function f coincides with the geometric density f n of µ, determined by Proposition 5.1.
Let us begin with some considerations preliminary to the proof. As µ is rigid motion invariant and monotone, its geometric densities f i , i = 0, . . . , n are defined (see Proposition 5.1). On the other hand, by Proposition 5.5, the only non-zero geometric density of µ is f n . Recall that f n is a non-negative decreasing function defined on R, moreover, as µ is m-continuous f n is right-continuous. Let f : (−∞, +∞] → [0, +∞) be the extension of f n , with the additional condition f (∞) := 0.
We will need the following Lemma.

Lemma 8.2.
Assume that µ is as in Theorem 8.1 and let f be the extension of its geometric density defined as above. Let K be a convex body and P = {K 1 , . . . , K N } be an inductive partition of K.
Let u ∈ C n such that L = dom(u) is a convex body, the restriction of u to L is continuous and int(L) ⊆ K. Then Proof. First we prove that u attains a maximum and a minimum when restricted to K i . Since u is lower semi-continuous and K i is compact and non-empty, we have inf K i u = min K i u. Suppose now that there exists a point x ∈ K i such that u(x) = ∞: in this case sup K i u = ∞ is attained at x; on the other hand, if K i ⊆ L, as the restriction of u to L (and thus to K i ) is continuous, Therefore, as f is decreasing, Using the monotonicity of µ and the definition of geometric densities we obtain Then This equality and (8.26) conclude the proof.
We will also need a well known theorem relating Riemann integrability and Lebesgue measure (known as Lebesgue-Vitali theorem). The proof of this result in the one-dimensional case can be found in standard texts of real analysis. The reader interested in the proof for general dimension may consult [1]. Theorem 8.3. Let g : R n → R be a bounded function which vanishes outside a compact set. Then g is Riemann integrable in R n if and only if the set of discontinuities of g has Lebesgue measure zero.
Proof of Theorem 8.1. Let us consider an arbitrary u ∈ C n which, from now on will be fixed. If dim(dom(u)) < n then f (u(x)) = 0 for H n -a.e. x ∈ R n ; hence, as µ is simple, i.e. the theorem is proven. Therefore in the remaining part of the proof we will assume that dim(dom(u)) = n.
Initially, we will assume that dom(u) = L is a convex body (with non-empty interior) and that the restriction of u to L is continuous. This implies in particular that g := f • u has compact support. We claim that the function g is Riemann integrable on R n . This will follow from Theorem 8.3 if we show that g = f • u is bounded on R n and the set of its discontinuities is a Lebesgue-null set.
It is easy to prove that g is bounded, since it is non-negative by construction and, as f is monotone decreasing, max R n (g) = f (min R n (u)) < ∞.
Since f is monotone decreasing, the set of its discontinuities is countable: let us call it {ξ i } i∈N and set ξ 0 = ∞. We claim that the set of discontinuities of g is contained in Let C ⊆ R n denote the set of points where g is continuous. We therefore aim to prove the following: which in turn is equivalent to Let us take a fixed x ∈ A; for every choice of i ≥ 0 there are two possibilities: Suppose (b) holds for two integers i, j; then Which means that (b) can happen at most once for every choice of x. Let us prove x ∈ C in case (b) never holds. As x / ∈ cl(u −1 (ξ 0 )), and ξ 0 = ∞, x is an interior point of the domain of u, so that u is continuous at x. Moreover, for every i ≥ 0 we have x / ∈ cl(u −1 (ξ i )) which implies x / ∈ u −1 (ξ i ), i.e. f is continuous at u(x). It follows that g is continuous at x. It remains to prove x ∈ C in case (b) holds for a specific j ≥ 0. Since which means u(B) = {ξ j }. Thus u, and also g, is constant on a neighborhood of x and hence continuous at x.
Having finally proven A ⊆ C it remains to show that A is a Lebesgue-null set. Since ∂u −1 (ξ i ) is the boundary of a convex body (possibly being empty) for every i, all these sets must be null sets, and so is their countable union (and therefore C itself is a null set, being a subset of a null set). Now let R be a closed rectangle such that int(L) ⊂ R; in particular g vanishes in the complement of R. As f is Riemann integrable, for every > 0 there exists a rectangular partition (see section 7.2.1) of R

and, clearly,
Without loss of generality we could assume that P is an inductive partition and thus, by (note that u + I R = u, as R contains the domain of u). Hence and, as is arbitrary, i.e. (8.24) is true for every u ∈ C n such that the domain of u is a compact convex set and u is continuous on its domain. To prove the same equality for a general u, let L i , i ∈ N, be an increasing sequence of convex bodies such that and consider the sequence of functions u i , i ∈ N, defined by u i = u + I L i . This is a decreasing sequence of elements of C n converging point-wise to u in the interior of dom(u). By the m-continuity of µ we have where, in the second equality, we have used the first part of the proof. On the other hand the sequence of functions f • u i , i ∈ N, is increasing and converges point-wise to f • u in R n . Hence, by the monotone convergence theorem, The proof of (8.24) is complete. As for (8.25), it follows from Proposition 6.5.
Remark 8.4. It is clear from the previous proof that the representation formula (8.24) of Theorem 8.1 remains valid for those functions u ∈ C n , such that: dom(u) = L ∈ K n and the restriction of u to L is continuous, even if we drop the assumption of m-continuity of µ.

Part one: n-homogeneous valuations
The following result is a direct consequence of Theorem 8.1 and Proposition 5.6. Theorem 9.1. Let µ : C n → R be a valuation with the following properties: • µ is rigid motion invariant; • µ is monotone decreasing; • µ is n-homogeneous; • µ is m-continuous.
Then there exists a function f : R → R, coinciding with the geometric density f n of µ, non-negative, decreasing, right-continuous and verifying the integrability condition where ν is the Radon measure related to f by Definition 9.2. (Extensions and restrictions of convex functions). Let k < n. Let u ∈ C k . We can now extend u to the whole R n in a canonical way by assigning the value ∞ where u was otherwise undefined: If u ∈ C k , then, it can be shown that u| n ∈ C n . On the other hand, the so-called restriction of a convex function u ∈ C n can be defined in the following way: It is immediate to show that u| k belongs to C k for every choice of u ∈ C n .

Definition 9.3. (Restrictions of valuations)
. Let k < n as above. Let µ be a real valuation on C n , then we can define the restriction of µ to C k as It is easy to verify that µ| k defined as above is a valuation on C k . Moreover, the valuation µ| k inherits the following properties from µ: rigid motion invariance, monotonicity, m-continuity and homogeneity. Let us now consider a valuation µ on C n and a convex function u ∈ C n such that dom(u) ⊆ {(x 1 , . . . , x n ) ∈ R n : x k+1 = · · · = x n = 0}. (9.27) Under these assumptions we have that The previous equality is an immediate consequence of Definition 9.3 and of the following consideration: (u| k )| n = u for every u ∈ C n which satisfies (9.27). Restricted valuations also share geometric densities up to the suitable dimension.
To be more precise, if f 0 , . . . , f n are the geometric densities of µ, then f 0 , . . . , f k are the geomtric densities of µ| k . To prove this, let t be a real number and H an arbitrary convex body in K k . Then, in other words, the function t + I K×P satisfies (9.27). We deduce that (9.28) holds for u = t + I K×P , thus A simple calculation yields Therefore we conclude by the arbitrariness of t and K.
The following corollary of Theorem 9.1 will be important in the sequel.
Let f k denote the k-th geometric density of µ. Then f k verifies the integrability condition Moreover, for every u ∈ C n such that dim(dom(u)) ≤ k where ν is a Radon measure on R and f k and ν are related by the identity Proof. Starting from µ we define its restriction to C k , µ| k . As remarked, µ| k is a valuation on C k with the following properties: it is rigid motion invariant, monotone decreasing, m-continuous and k-homogeneous. Denote by g i , i ∈ {0, 1, . . . , k} its geometric densities; then g 0 ≡ . . . g k−1 ≡ 0 and g k = f k . By Theorem 9.1 we have that f k verifies the claimed integrability condition and for every v ∈ C k , where ν and f k are related as usual by f k (t) = ν((t, ∞)). Now let u ∈ C n be such that dim(dom(u)) ≤ k; we want to compute µ(u). As µ is rigid motion invariant, without loss of generality we may assume that dom(u) ⊆ {(x 1 , . . . , x n ) ∈ R n : x k+1 = · · · = x n = 0}.
Then there exists a Radon measure ν defined on R, verifying the integrability assumption for every u ∈ C n . Moreover, the measure ν is determined by the unique non-vanishing geometric density f k of µ as follows: The rest of this section is devoted to the proof of this result; throughout, µ will be a valuation with the properties indicated in the previous theorem. Note that the validity of the Theorem for k = n is established by Theorem 9.1.
By Proposition 5.1 we may assign to µ its geometric densities f j , j = 0, . . . , n. By homogeneity we have f j ≡ 0 for every j = k. In other words, the only density which can be non-identically zero is f k . For simplicity we will call this function f . By the properties of µ, this is a non-negative decreasing function; moreover, as µ is m-continuous, f is right-continuous on R and, by Corollary 9.4, it verifies the integrability condition We proceed by induction on the dimension n. Let us then start from the case n = 1. As the theorem is already proven for k = n = 1 we only need to consider the case k = 0; but this follows from Proposition 5.4: Theorem 9.5 is proven in dimension n = 1.
To continue with the induction argument, we assume that the theorem holds up to dimension (n − 1) and we are going to prove it in the n-dimensional case. We may assume that 1 ≤ k ≤ n − 1.
In the next part of the proof we will assume, in addition to the above properties, that the only non-zero density f of µ is smooth: f ∈ C ∞ (R). As in the previous sections, we introduce the Radon measure ν related to f by the identity Based on ν, we construct an auxiliary valuation µ a : C n → R defined as follows µ a (u) = R V k (cl({u < t}))dν(t).
By the results of section 5, µ is a well defined valuation and it is rigid motion invariant, decreasing, k-homogeneous and m-continuous. Moreover, its geometric density of order k is precisely f , i.e. the same as µ.
The idea is to prove that µ r is identically zero. Note that µ r inherits most of the properties of µ and µ a : it is a valuation, rigid motion invariant, k-homogeneous and m-continuous. We cannot infer in general that µ r is monotone.
Claim 1. The valuation µ r "vanishes horizontally", i.e. for every convex body K ∈ K n and every t ∈ R we have µ r (t + I K ) = 0.
The proof is a straightforward consequence of the fact that µ and µ a have the same geometric densities.
Claim 2. The valuation µ r is simple.
Proof. Let u ∈ C n be a convex function whose domain has dimension strictly less than n. As µ r is rigid motion invariant, we might assume without loss of generality that u ⊆ {(x 1 , . . . , x n ) ∈ R n : x n = 0}.
We will now introduce a construction which is going to help us evaluate a valuation on piece-wise linear functions. We fix e ∈ R n s.t. |e| = 1, p ≥ 0, V = pe.
Let µ 0 be a valuation on C n and consider the linear function w : R n → R defined by Then we define a mapping on the family of convex bodies of R n , σ µ 0 ,V : K n → R, as follows It is easy to check that σ µ 0 ,V is a valuation. From now on throughout this paper, we will consider the previous construction specialized to valuations which are rigid motion invariant, hence we will assume without loss of generality that V = pe n = (0, . . . , 0, p) ∈ R n , p ≥ 0, and consequently that w(x) = w p (x) = p(x, e n ) = px n for all x ∈ R n . Moreover, for the sake of brevity, we will introduce the following simplified notation: The following claim collects some of the properties of σ r that will be used in the sequel.
Claim 3. σ r has the following properties: 1. it is a valuation on K n ; 2. it is simple; 3. it is invariant with respect to every rigid motion T of R n such that where T is a rigid motion of R n−1 .
Proof. Let K, L ∈ K n be such that K ∪ L ∈ K n . Then These relations remain valid if we add w as follows w + I K∪L = (w + I K ) ∧ (w + I L ), w + I K∩L = (w + I K ) ∨ (w + I L ).
We conclude that σ r is a valuation. If K ∈ K n has no interior point, the domain of I K , and consequently that of w + I K , have the same property. Then, σ r (K) = µ r (w + I K ) = 0.
We anticipate that the following step is one of the most delicate in the proof.
Proof. We first treat the easier case p = 0, which leads to w ≡ 0 so that where we have used Claim 1. Next we assume p > 0. Given two real numbers α, β with α ≤ β, we define the strip: Let K ∈ K n and let y m and y M be such that the hyperplanes of equations x n = y m and x n = y M are the supporting hyperplanes to K with outer unit normals −e n and e n respectively. In other words, Hence where, for the second term we have used the equality, due to the condition k ≥ 1 and the integrability condition on f , lim t→∞ f (t) = 0.
Consequently we have the following bounds: Note that the function is Lipschitz continuous in a neighborhood of τ = 0 (indeed, as already remarked before, its 1/k power is concave in [y m , y M ]) and, by monotonicity of intrinsic volumes, it is bounded by V k (K), i.e. a constant independent of y and h. Then, as f is smooth, it follows from (9.32) and (9.33) that φ is Lipschitz continuous in [y m , y M ]; in particular (9.33) implies that φ (y) ≥ 0 for every y for which φ is defined. As φ(y m ) = σ r (K ∩ S[y m , y m ]) = 0 (recall that σ r is simple), we have that The proof is complete.
For the sequel we will need the following result (which could be well-known in the theory of valuations on convex bodies). Lemma 9.6. Let σ : K n → R be a valuation which is non-negative and simple. Then σ is monotone increasing on the class of polytopes, i.e. for every P and Q polytopes in R n such that P ⊆ Q we have σ(P ) ≤ σ(Q).
Proof. Let P and Q be polytopes such that P ⊆ Q; let F be a family of hyperplanes in R n , defined as follows: F(P, Q) = {H is a hyperplane containing a facet of P and H ∩ int(Q) = ∅} , and let N (P, Q) ≥ 0 be the cardinality of F(P, Q). We will prove that by induction on N (P, Q). If N (P, Q) = 0 we have that P = Q so that there is nothing to prove. Assume that the claim is true up to (n − 1), for some n ∈ N, and that N (P, Q) = n. Let H ∈ F and let H + and H − be the closed half-spaces determined by H. We may assume that P ⊆ H + . Let Q + = Q ∩ H + and Q − = Q ∩ H − (which are still polytopes); as and as σ is simple and non-negative On the other hand Q + ⊇ P and F(P, Q + ) ⊂ F(P, Q), in particular N (P, Q + ) < N (P, Q) so that, by the induction assumption, Claim 5. The valuation σ r is monotone increasing.
Proof. Let K, L ∈ K n be such that K ⊆ L; there exist two sequences of polytopes P i , Q i , i ∈ N, with the following properties: 1. they are increasing with respect to set inclusion; 2. P i → K and Q i → L as i tends to infinity, in the Hausdorff metric; 3. P i ⊆ Q i for every i ∈ N.
In particular σ r (P i ) ≤ σ r (Q i ). Now, recalling the definition of σ r we have that Note that u i is a decreasing sequence, and it converges point-wise to u : R n → R defined by in the relative interior of K. As µ a and µ are m-continuous we have In a similar way we can prove that Since, as already pointed out, σ r (P i ) ≤ σ r (Q i ) for all i ∈ N, passing to the limit for i → ∞ yields the claimed σ r (K) ≤ σ r (L).
Let us make a further step to investigate the behavior of µ r on restrictions of linear functions. Given a function u ∈ C n−1 , we may consider the set Claim 6. Let p > 0. For every u ∈ C n−1 the function w + I epi(u) belongs to C n .
Proof. The set epi(u) is convex and closed (by the semi-continuity of u). Hence the function v = w + I epi(u) is lower semi-continuous and convex. Let x i , i ∈ N, be a sequence in R n such that lim From any subsequence of x i we may extract a further subsequence (let us call itx i ) such that eitherx i ∈ epi(u) for every i orx i ∈ R n \ epi(u) for every i. In the second case we have v(x i ) = ∞ for every i. In the first case we have, Proposition 3.8). As |x i | is unbounded, we must have thatȳ i is not bounded from above and, up to extracting a further subsequence we may assume that This implies that v(x i ) = py i tends to infinity as well. Hence from any subsequence x i such that |x i | → ∞ we may extract a subsequencex i such that v(x i ) tends to infinity. Hence lim |x|→∞ v(x) = ∞ and we conclude that v ∈ C n .

Claim 7.
Let p > 0. The applicationμ has the following properties: 1. it is a rigid motion invariant valuation; 2. it is simple; 3. it is monotone decreasing.
Proof. We will denote a point in R n by (x , y), with x ∈ R n−1 and y ∈ R. For u ∈ C n−1 and t ∈ R set By the m-continuity of µ r , µ(u) = lim t→∞ µ r (w + I epi t (u) ) = lim t→∞ σ r (epi t (u)).
We will see that properties 1 -3 follow easily from this characterization ofμ and Claim 3. Assume that T is a rigid motion of R n−1 . Definē T is a rigid motion of R n and it verifies (T (x), V ) = (x, V ) for every x ∈ R n . Then, by item 3 in Claim 3, σ r (epi t (u)) = σ r (T (epi t (u))).
Replacing this equality in the previous one, and letting t → ∞, we get which proves thatμ is rigid motion invariant. To prove thatμ is simple, let u ∈ C n−1 be such that dim(dom(u)) ≤ (n − 2). Then dim(epi(u)) ≤ (n − 1) and As µ r is simple, σ r is simple, by Claim 3. Hence σ r (epi t (u)) = 0 ∀ t.
As for monotonicity, if u and v belong to C n−1 and are such that u ≤ v in R n−1 , then As σ r is monotone increasing we get The conclusion follows letting t to ∞.
Claim 8. Let p > 0. There exists a functionf : R × (0, ∞),f =f (t, p), such that for every u ∈ C n−1 such that dom(u) ∈ K n and the restriction of u to dom(u) is continuous (here dx denotes the usual integration in R n−1 ).
Proof. By Claim 7 we may apply Theorem 8.1 and subsequent Remark 8.4 to deduce (9.34).
Given K ∈ K n−1 and t 1 , t 2 ∈ R, with t 1 ≤ t 2 , we consider the cylinder: Evaluating σ r on cylinders is a crucial step, as we will see in the following claim.
Claim 9. Let p > 0. For every K ∈ K n−1 , and every t 1 , t 2 ∈ R with t 1 ≤ t 2 we have: Proof. We have, for every t 1 ∈ R, µ r (w + I K×[t 1 ,∞) ) = µ r (w + I epi(u) ) (9.36) On the other hand, for t 1 , t 2 ∈ R with t 1 ≤ t 2 we have Hence, as µ r is a valuation and it is simple, and as dim(K × {t 2 }) ≤ n − 1, we obtain The next step is to deduce further information aboutf exploiting the homogeneity of µ r (recall that µ r is homogeneous of order k).

Claim 10.
There exists a non-negative decreasing function φ : R → R such that Proof. We recall that w p (x) = p(x, e n ) for every choice of p ≥ 0 and x ∈ R n . As before, let K ∈ K n−1 and let λ > 0; we have, for x ∈ R n and t ∈ R, By the homogeneity of µ r and µ r (w p/λ + I λK×[λt,∞) ) = V n−1 (λK)f λt, p λ = λ n−1 V n−1 (K)f λt, p λ by the homogeneity of intrinsic volumes. Hence, as we may chose K so that V n−1 (K) > 0, we obtain that for every t ∈ R, p > 0 and λ > 0 we havē where we have set φ(s) =f (s, 1) for all real s. Asf is non-negative and decreasing with respect to t for every p > 0 the claim follows.
In the next step we prove that the m-continuity of µ r implies that the function φ is constant.
Claim 11. The function φ introduced in the previous step is constant in R (in particular σ r vanishes on cylinders).
Proof. By the previous steps we have that for every K ∈ K n−1 and for every t 1 , t 2 ∈ R with t 1 ≤ t 2 , µ r (w + I K×[t 1 ,t 2 ] ) = V n−1 (K)(φ(pt 1 ) − φ(pt 2 )) p j . (9.37) Let K be the (n − 1)-dimensional unit cube with centre at the origin and let We also set, for i ∈ N, In particular v i is a decreasing sequence of functions in C n converging to I D in the relative interior of D, so that by m-continuity we have where we have used the fact that µ r vanishes horizontally (Claim 1). We may also write and using the fact that µ r is a simple valuation, and Claim 1 again, we get that On the other hand, by translation invariance, if we set where w i (x) = si x n and t i = 1 i .
Consequently, by (9.37) Letting i tend to infinity this quantity must tend to zero, by the previous part of the proof; as j ≥ 0 and V (K) > 0, the only possibility is φ(s) = φ(0). This proves that φ is constantly equal to φ(0) in [0, ∞).
To achieve the same result in (−∞, 0] we may argue in a similar way. Let s < 0 and K, D, E i as above. Setv i (x 1 , . . . , x n ) = −si x n and v i = (I D + s) ∨v i . This is again a decreasing sequence in C n , converging to s + I D in the relative interior of D; by Claim 1: lim On the other hand The conclusion φ(s) = φ(0) follows as above.
Claim 12. Let V ∈ R n , c ∈ R and K ∈ K n ; define Then µ r (u) = 0.
Proof. Assume first that c = 0. If V = 0 the assert follows from Claim 1. Assume V = 0 and let V = pe, with p > 0 and e a unit vector. Recalling the definition of σ µr,V we have: On the other hand, since µ r is rigid motion invariant, we can assume, without loss of generality, that V = e n and, as remarked in Claim 11, setting as before w : R n → R defined by w(x) = (x, pe n ), we get for every H ∈ K n−1 , t 1 , t 2 ∈ R such that t 1 ≤ t 2 . Let us choose H, t 1 and t 2 such that Then, as σ µr,V is non-negative and monotone increasing (Claims 4 and 5), The case c = 0 is readily recovered by the previous one using the translation invariance of µ r .
The last result will open the way to prove that µ r vanishes on piece-wise linear functions and, eventually, it vanishes identically on C n . Definition 9.7. A function u ∈ C n is said to be piece-wise linear if: • dom(u) = P is a polytope; • there exists a polytopal partition P = {P 1 , . . . , P N } of P such that for every i ∈ {1, . . . , N } there exists V i ∈ R n and c i ∈ R such that Claim 13. The valuation µ r vanishes on piece-wise linear functions.
Proof. As any polytopal partition admits a refinement which is a complete partition (see Remark 7.4 in section 7.2), without loss of generality we may assume that P is complete, so that in particular it is an inductive partition (see Proposition 7.5). The claim follows immediately from Claim 12, the fact that µ r is simple, and Lemma 7.6.

Claim 14.
The valuation µ r vanishes on C n .
Proof. Let u ∈ C n ; if the dimension of dom(u) is strictly less than n, µ r (u) = 0 as µ r is simple. So, assume that Ω = int(dom(u)) = ∅. Let P be a polytope contained in Ω, and let u i , i ∈ N, be a sequence of piece-wise linear functions of C n , such that for every i: dom(u i ) = P , u i ≥ u i+1 in P , and the sequence u i converges uniformly to u in P ; such a sequence exists by standard approximation results of convex functions by piece-wise linear functions. Using the m-continuity of µ r and the previous Claim 13, we obtain Now take a sequence of polytopes P i , i ∈ N, such that: P i ⊆ P i+1 ⊆ Ω for every i and Ω = i∈N P i .
Then the sequence is formed by elements of C n , is decreasing, and converges point-wise to u in Ω; by mcontinuity and the previous part of this proof The proof of Theorem 9.5 is complete, under the additional assumption that the density f of µ is smooth. The next and final step explains how to deduce the theorem in the general case.

Claim 15.
The assumption that f is smooth can be removed.
Proof. Let µ be as in the statement of Theorem 9.5, and let µ i , i ∈ N, be the sequence of valuations determined by Proposition 5.7 (taking for example = 1/i, i ∈ N). It follows from the definition of µ i given in section 5.1 that, as µ is k-homogeneous, µ i is k-homogeneous as well. Moreover, the only non-vanishing geometric density of µ i , that we will denote by f i , is smooth. Hence, for every i we may apply the previous part of the proof to µ i and deduce that where ν i is a Radon measure on R and it is related to f i by the equality We apply Proposition 6.4 to get we recall that β k (u; ·) is the distributional derivative of the increasing function R t → V k (cl({u < t})).
From Proposition 3.4 we know that β k (u; ·) can be decomposed as the sum of a part which is absolutely continuous with respect to the one-dimensional Lebesgue measure and a Dirac point-mass measure having support at m(u) and weight V k ({x : u(x) = m(u)}). If in particular we assume that u ∈ C n is such that {x : u(x) = m(u)} consists of a single point, (9.38) we have that (as k ≥ 1) V k ({u = m(u)}) = 0, so that β k (u; ·) is absolutely continuous with respect to the Lebesgue measure on the real line. Our next move is to prove that, under the assumption (9.38) We know that the sequence f i converges to f almost everywhere on R with respect to the Lebesgue measure, and hence with respect to β k (u; ·). Note also that f i (t) = R f (t − s)g 1/i (s)ds = As f is decreasing (and non-negative) On the other hand R f (t − 1)dβ k (u; t) = R f (t)dβ k (u; t + 1) = R f (t)dβ k (ū; t) = R V k (cl({ū < t}))dν(t) < ∞, whereū = u − 1 and the last inequality is due to the integrability condition on f (Proposition 6.1 and Corollary 9.4). Hence we may apply the dominated convergence theorem and obtain (9.39). Note that if u verifies condition (9.38), then so does the function u + s, for every s ∈ R. By Proposition 5.7 we conclude that µ(u+s) = R f (t)dβ k (u+s; t) = R f (t)dβ k (u; t−s) = R f (t+s)dβ k (u; t), for a.e. s ∈ R.
(9.40) Let s i , i ∈ N, be a decreasing sequence of real numbers converging to zero such that (9.40) holds true; then by m continuity of µ lim i→∞ µ(u + s i ) = µ(u).
The m-continuity implies also that f is right-continuous (see right after Corollary 6.3), hence lim i→∞ f (t + s i ) = f (t) ∀ t ∈ R.
Using again the monotonicity of f and the monotone convergence theorem we obtain lim i→∞ R f (t + s i )dβ k (u; t) = R f (t)dβ k (u; t).
Putting the last equalities together we arrive to for every u ∈ C n verifying (9.38). The last step will be to prove that this equality is true for every u ∈ C n . For i ∈ N set u i : R n → R u i (x) = u(x) + |x| 2 i .
Clearly u i ∈ C n and, as u i is strictly convex it verifies condition (9.38) and, consequently, (9.41). By m-continuity lim i→∞ µ(u i ) = µ(u).
We need to prove that lim i→∞ R V k (cl({u i < t}))dν(t) = R V k (cl({u < t}))dν(t). (9.42) As u i ≥ u in R n fo every i we have that {u i < t} ⊆ {u < t} for every t. We have already proven that lim i→∞ cl({u i < t}) = cl({u < t}) ∀ t ∈ R, where the limit is intended in the Hausdorff metric on K n . Then (9.42) follows by the monotone the monotone convergence theorem, and Theorem 9.5 is finally proven in the general case as well.

A non level-based valuation
In this section we will present a way to construct monotone valuations on C n which are moreover rigid motion invariant and m-continuous and, despite verifying all these desirable properties, cannot be expressed as a linear combination of homogeneous valuations on C n . Fix n, m ∈ N, for all u ∈ C n we setû(x, y) = u(x) + |y| for all (x, y) ∈ R n × R m , where | · | is to be interpreted as the Euclidean norm in R m . Note that if u ∈ C n , then u ∈ C n+m .
We are now ready to define the prototype of the valuations described at the beginning of this section. Proof. First of all, for ease of notation, set µ(·) = V k (cl({· < t})). By Proposition 6.2, µ verifies all the properties i) -iv).
i) As a preliminary step to prove the condition on ∞, notice that ∞ = ∞ ∈ C n+m . As a matter of fact, for all (x, y) ∈ R n × R m we have ∞(x, y) = ∞(x) + |y| = ∞ + |y| = ∞.
Let now u, v ∈ C n , we have Let us specialize the valuation of Proposition 10.1 to the case n = m = k = 1. In this case we can provide a simple geometric explanation: V 1 (cl({û < t})) is equal to the length of that portion of the graph of u that lies strictly under the level t and therefore we will refer to it as undergraph-length.
In other words, cl({û < t}) be obtained as a result of the following process: take the part of epi(u) that lies below the line {(x, y) ∈ R 2 : y = t}, translate it "vertically" so that the flat top is now lying on the x-axis H := {(x, 0) ∈ R 2 }, finally symmetrize it with respect to H. We recall that V 1 (K) coincides with the length (1-dimensional Lebesguemeasure) in case dim(K) = 1 and with 1 2 H 1 (∂K) when dim(K) = 2 (see [16]). If dom(u) has dimension 1, as t > m(u), the epi(u) ∩ {(x, y) ∈ R 2 : y ≤ t} is 2-dimensional and therefore V 1 (cl({û < t})) is equal to the length of the graph of u that lies strictly under the level t for every choice of t ∈ R.
The undergraph-length is not a level based valuation. By these words we mean that we could actually take a convex function u ∈ C 1 , rearrange its levels using translations and obtain another convex function v such that V 1 (cl({û < t})) = V 1 (cl({v < t})) for all t > m(u). Take for instance u(x) = |x| and v(x) = x/2 + I [0,∞) for all x ∈ R; we have {u < t} = (−t, t) and {v < t} = (0, 2t) = t + (−t, t) for all positive real t. On the other hand, their undergraph-lengths differ: a quick use of the Pythagorean theorem reveals that V 1 (cl({û < t})) = 2 √ 2t while V 1 (cl({v < t})) = √ 5t. The length of the undergraph is a valuation which is completely different from the ones we have studied so far: not only it is not α-homogeneous for any real α, it turns out that V 1 (cl({ · < t})) cannot even be written as a finite sum of homogeneous functions. To prove this consider the following u ∈ C 1 , defined as u(x) = |x| for all x ∈ R. For all λ > 0 we get V 1 (cl({ u λ < 1})) = 2 √ 1 + λ 2 . Since V 1 (cl({ u λ < 1})) is not a polynomial in λ, V 1 (cl({ u λ < 1})) cannot be decomposed into the (finite) sum of homogeneous functions. This implicitly tells us that under these assumptions (monotonicity, rigid motion invariance and m-continuity), homogeneous valuations do not form a basis for the vector space of valuations on C n .