On the monotonicity of weighted perimeters of convex bodies

We prove that, among weighted isotropic perimeters, only constant multiples of the Euclidean perimeter satisfy the monotonicity property on nested convex bodies. Although the analogous result fails for general weighted anisotropic perimeters, a similar characterization holds for radially-weighted anisotropic densities.

1. Introduction 1.1.Monotonicity property.Let N ≥ 2. If A, B ⊂ R N are two nested convex bodies, that is compact convex sets with non-empty interior such that A ⊂ B, then where P (E) = H N −1 (∂E) denotes the Euclidean perimeter of the convex body E ⊂ R N .The monotonicity property (1.1) is well known and dates back to the ancient Greeks (Archimedes took it as a postulate in his work on the sphere and the cylinder [1, p. 36]).Inequality (1.1) can be proved in several ways: by the Cauchy formula for the area surface of convex bodies [5, §7]; by the monotonicity property of mixed volumes [5, §8]; by the Lipschitz property of the projection on a convex closed set [6,Lem. 2.4]; by the fact that the perimeter is decreased under intersection with half-spaces [25,Ex. 15.13].
Inequality (1.1) extends to the anisotropic (Wulff ) Φ-perimeter where ν E : ∂E → S N −1 is the inner unit normal of the convex body E ⊂ R N (defined H N −1 -a.e. on ∂E) and Φ : R N → [0, +∞] is a fixed lower-semicontinuous, positively 1homogeneous and convex function.Clearly, If Φ = | • |, then P Φ (E) = P (E).Similarly to (1.1), the monotonicity of the Φ-perimeter is a consequence of one of the following: the Cauchy formula for the anisotropic perimeter [5, §7]; the monotonicity property of mixed volumes [5, §8]; the fact that the anisotropic perimeter is decreased under intersection with half-spaces [25,Rem. 20.3].
In passing, we mention that the monotonicity property holds even for perimeter functionals of non-local type, as the fractional perimeter [16,Lem. B.1] and, more generally, non-local perimeters induced by a suitable interaction kernel [3,Cor. 2.40].
The monotonicity property of perimeters has gained increasing attention in recent years.We refer to [7,8,24,31] and the survey [20] for quantitative versions of the monotonicity inequality (also see [22] for the quantitative monotonicity in the non-local setting), and to [4,9,12,15,21,23,26] for some applications and related results.
1.2.Main result.In this note, we are interested in studying the monotonicity property on nested convex bodies for the class of weighted perimeters.Given a Borel function f : R N → [0, +∞], we let be the weighted (isotropic) perimeter of the convex body E ⊂ R N .Clearly, if f ≡ c for some c ∈ [0, +∞), then P f = c P , a constant multiple of the Euclidean perimeter.Weighted perimeters have been largely investigated in relation to isoperimetric, cluster and Cheeger problems, see [2,10,11,13,[17][18][19][28][29][30] and the survey [27] for an account on the existing literature.
Our main result is the following rigidity property, namely, the only weighted perimeter satisfying the monotonicity property is (a constant multiple of) the Euclidean perimeter.then f ≡ c a.e. for some c ≥ 0.
Theorem 1.1 is quite intuitive.In fact, one clearly expects that, if f is not constant in some direction, then the monotonicity property should be violated on any suitable family of convex bodies with some side (continuously) deforming along that direction.However, one should carefully keep into account the values of f on the entire boundary of each convex body of the family, which forces one to consider deformations in that direction given by graphs of concave functions fixing the boundary of the chosen side.
One may wonder whether the analog of Theorem 1.1 holds for weighted anisotropic perimeters.More precisely, given a non-negative Finslerian weight f : R N × S N −1 → [0, +∞] (i.e., possibly depending also on the inner unit normal ν E : ∂E → S N −1 of the convex body E ⊂ R N ) and assuming the monotonicity of the weighted anisotropic perimeter P f , is it true that f = f (x, ν) does not depend on x?This is in general false.As a counterexample, consider any bounded vector field F ∈ C 1 (R N ; R N ) with constant divergence, div F ≡ α for some α ∈ [0, +∞), and define the anisotropic weight where β ∈ [ F ∞ , +∞) ensures the non-negativity of the weight f .By the Divergence Theorem, the anisotropic weighted perimeter readily yielding the desired monotonicity property in virtue of that of the Euclidean perimeter (1.1) and that of the Lebesgue measure with respect to nestedness.Despite the counterexample in (1.4), from Theorem 1.1 we can deduce the following result, which provides a partial analog of the rigidity property in the anisotropic regime under some additional structural assumptions on the weight function.
The proof of Corollary 1.2 combines the invariance of the monotonicity property with respect to rotations with Theorem 1.1.

Proofs of the statements
2.1.Proof of Theorem 1.1.We begin by observing that it is not restrictive to assume that f ∈ C ∞ (R N ).Indeed, given A ⊂ B two nested convex bodies in R N , the translated sets A + y ⊂ B + y are still two nested convex bodies for any y ∈ R N .Therefore, in virtue of (1.3) and changing variables, we get (2.1) Now fixed any family of non-negative mollifiers (̺ ε ) ε>0 ⊂ C ∞ c (R N ), multiplying (2.1) by ̺ ε (y), integrating on R N with respect to y, and owing to Tonelli's Theorem, we infer that By the arbitrariness of the nested convex bodies A and B, the weight 3) for each ε > 0. In particular, since f ε converges to f in L 1 loc , if we show that ∇f ε ≡ 0 for each ε > 0, then f ≡ c a.e. for some c ≥ 0.
Consequently, from now on, we assume that f ∈ C ∞ (R N ).We now claim that ∂ x N f (x) = 0 for each x ∈ R N .By the translation invariance in (2.1), we just need to show that ∂ x N f (0) = 0.
Let δ > 0 to be chosen later on.For λ ∈ R, we define Note that E(λ) and Hence, in virtue of (1.3), we get that By the area formula, the above inequality rewrites as In particular, the function ℓ : R → [0, +∞), given by achieves its minimum at λ = 0, so that ℓ ′ (0) = 0. Being f smooth, we can exchange the differentiation and the integration signs, obtaining In particular, h(x ′ ) > 0 for all x ′ ∈ (−δ, δ) N −1 as soon as h ≡ 0. By contradiction, if ∂ x N f (0) = 0, then, by smoothness of f , we may assume that ∂ N f (x) has constant sign for each x ∈ B r (0) for some r > 0. Choosing δ > 0 so small that [−δ, δ] N −1 × {0} ⊂ B r (0), the equality (2.2) immediately yields a contradiction.
In the previous argument, the choice of fixing the N-th component does not play any role and can be repeated almost verbatim to show that ∂ x i f (0) = 0 for each i = 1, . . ., N. Thus, again by the translation invariance (2.1), we get that ∇f (x) = 0 for all x ∈ R N , yielding the conclusion.Remark 2.1.In the above proof, one needs much less than the monotonicity of the perimeter on nested convex sets in order to conclude that the weight is constant.Indeed, it would be enough to know that, for each direction e i ∈ S N −1 , i = 1, . . ., N, and each point x ∈ R N , the monotonicity property holds on two hypercubes (not necessarily with the same edge size) with a face containing x and orthogonal to e i with opposite outward normals ±e i on that face.

Proof of Corollary 1.2.
Let us denote by SO(N) be the special orthogonal group, and let µ ∈ P(SO(N)) be the (unique) Haar probability measure on SO(N) (see [14] for a detailed exposition).Given A ⊂ B two nested convex bodies in R N , the rotated sets R(A) ⊂ R(B) are still two nested convex bodies for any R ∈ SO(N).Therefore, in virtue of (1.3) and changing variables, we get owing to the elementary facts that R(∂E) = ∂R(E) and that ν R for R ∈ SO(N).We now claim that the function is constant.Indeed, given any ν ∈ S N −1 , then we can find R ν ∈ SO(N) such that ν = R ν (e 1 ).Due to the invariance properties of the Haar measure µ, we can compute where, with a slight abuse of notation, Q → µ(QR −1 ν ) stands for the push-forward of the measure µ with respect to the right translation by R −1 ν .Hence, integrating on SO(N) with respect to µ, using Tonelli's Theorem, the above equality, that Φ > 0, and simplifying, from (2.4) we get Remark 2.2.One could slightly weaken the hypotheses of Corollary 1.2 by allowing Φ to also attain zero.In fact, it is enough to require that the integral in (2.5) is not zero.Email address: giorgio.stefani.math@gmail.comor gstefani@sissa.it

Theorem 1 . 1 .
Let f : R N → [0, +∞] be a Borel function such that f ∈ L 1 loc (R N ).If the weighted perimeter P f in (1.2) satisfies the monotonicity property, i.e., P f (A) ≤ P f (B) for any two nested convex bodies A ⊂ B in R N ,(1.3) ∂A g(x) dH N −1 (x) ≤ ∂B g(x) dH N −1 (x)for any two nested convex bodies A ⊂ B. The conclusion follows from Theorem 1.1.