Volumetric Properties of the Convex Hull of an n-dimensional Brownian Motion

Let K be the convex hull of the path of a standard brownian motion B(t) in R^n, taken at time 0<t<1. We derive formulas for the expected volume and surface area of K. Moreover, we show that in order to approximate K by a discrete version of K, namely by the convex hull of a random walk attained by taking B(t_n) at discrete (random) times, the number of steps that one should take in order for the volume of the difference to be relatively small is of order n^3. Next, we show that the distribution of facets of K is in some sense scale invariant: for any given family of simplices (satisfying some compactness condition), one expects to find in this family a constant number of facets of tK as t approaches infinity. Finally, we discuss some possible extensions of our methods and suggest some further research.


Introduction
Convex hulls seem to attract a significant amount of interest, in some cases for representing physical phenomena and in others for their central importance in many algorithmic methods. Random convex hulls, in numerous different settings, have been widely studied by probabilists and geometers (see [CMR] for and extensive survey). One example of a random convex hull that has been studied is the convex hull of a Brownian motion, which may represent, for example, the domain of influence on a diffusing particle in a certain physical system. The object of this paper is to further study the object generated by taking the convex hull of the standard Brownian motion in R n .
The convex hull of the path of the planar Brownian motion has been quite extensively studied. Much is known about this object, including its expected area and perimeter length, the degree of smoothness of its boundary, the rate of convergence of the area of a convex hull of a random walk to its area, etc. (see e.g. [CMR], [Bax], [BL], [T], [ElB], [CHM], [BL] and references therein). However, it seems like much less is known about the convex hull of the Brownian motion in higher dimensions. Two examples of notable works concerning the higher dimensional case are a paper by Kampf, Last and Molchanov, [KLM] in which, for instance, the first and second intrinsic volumes are calculated and a work by Kinney, [K], in which a bound for the total curvature is established.
We extend some known results from the planar case to the higher dimensional case, as well as obtain certain asymptotics of the behaviour of these objects as the dimension goes to infinity. We introduce new methods which may be further used to study volumetric and combinatorial properties of the convex hull of the Brownian motion and random walk.
Some of our methods are quite related to the ones used in the world of asymptotic convex geometry, the concentration of measure phenomena, for example. Since there are not many concrete examples of high-dimensional convex bodies, for a convex geometer the main interest in investigating these objects may be to provide new examples of high dimensional convex bodies, which may hopefully lead to a deeper understanding of the theory of convex bodies and may possibly also provide counter-examples to certain assumptions about convex bodies. The methods introduced in this work may be seen as a basic toolbox for the convex geometer to analyze these bodies.
Let us introduce our setting. Fix a dimension n ∈ N. Let B(·) be a standard Brownian motion in R n . For a subset A ⊂ R n , by conv(A) we denote the minimal convex set containing A, the convex hull of A. Our main object of concern in this paper will be K = conv({B(t), 0 ≤ t ≤ 1}).
We will be interested, for example, in its expected volume and in its distribution of facets. In order to better study these properties, in many cases it will be convenient to introduce an approximation for this object by a simpler object, namely the convex hull of a random walk. We construct the random walk as follows: Let P = ((x 1 , y 1 ), (x 2 , y 2 ), ...) be a Poisson point process of intensity 1 in the set [0, 1] × [0, ∞] and for all α ≥ 0, define, The process Λ can be thought of as a "Poisson rain" on the interval [0, 1]: note that for all α ≥ 0, Λ α is a Poisson point-process of intensity α on the unit interval and that the family Λ α is increasing with α. For a fixed value of α, writing Λ α = (t 1 , ..., t N ) where t 1 ≤ ... ≤ t N , we can think of (B(t 1 ), B(t 2 ), ..., B(t N )) as a random walk in R n . Finally, for all α > 0, we define so K α is a monotone sequence of discrete approximations of K, each defined as the convex hull of a certain random walk.
For a measurable set L ⊂ R n , we denote the k-dimensional Hausdorff measure of L by V ol k (L). By ∂L we denote the boundary of L. The first theorem we prove is a formula for the expected volume and surface area of K. The theorem reads, Theorem 1.1 One has, for every dimension n ≥ 2, E[V ol n (K)] = π 2 n/2 1 Γ(n/2 + 1) 2 .
Remark 1.1 It was pointed to us by Christoph Thäle that as a corollary of the above theorem, one may attain formulae for all intrinsic volumes of the body K, thus generalizing one of the results appearing in [KLM]. By using Kubota's formula in order to express the j-th intrinsic volume as the average of the volumes of j-dimensional projections of K with respect to the Haar measure on the corresponding Grassmanian, one can derive the formula E[V j (K)] = n j π 2 j/2 Γ((n − j)/2 + 1) Γ(j/2 + 1)Γ(n/2 + 1) where V j (K) is the j-th intrinsic volume of K.
Our next result is a derivation of asymptotics for the number of steps needed in order to approximate the convex hull of the Brownian motion, K, by the convex hull of the random walk, K α . Our theorem roughly states that the correct order of points needed in order for the volume of K α to be a proportion of the volume of K is n 3 . It reads, Theorem 1.2 One has the following bounds: For all n ≥ 2 and all α > 0, (1) On the other hand, for all α < n 3 /8, one has Note that, according to the above theorem, for any given proportion constant R < 1, there exists a constant C(R) independent of the dimension, such that whenever α > C(R)n 3 , the proportion between the expected volume K α out of the entire volume of K will be at least R. By basic properties of the Poisson process, the same will be true if we take C(R)n 3 uniform points on the interval [0, 1]. On the other hand, the second part of the theorem shows us that taking only o(n 3 ) points will yield E[V ol n (K α Our last result concerns with the distribution of facets of K. In order to formulate it, we need some notation. For two (n − 1)-dimensional simplices s 1 , s 2 ⊂ R n we say that s 1 and s 2 are equivalent if they are equal up to some translation. From this point further, by slight abuse of terminology, the term simplex will refer to an equivalence class of simplices. We denote by S the set of (n − 1)-dimensional simplices and let F (K) ⊂ S denote the set of (n − 1)dimensional facets of K (i.e., the set of (n − 1)-dimensional simplices lying entirely in the boundary of K, which are maximal in the sense that they are not strictly contained in any simplex lying in the boundary of K). For a family of simplices C ⊂ S, we define the expected number of facets of K which are in C. Our aim is to study the behaviour of M K (C).
Next, for a set L ⊂ R n and for ǫ > 0, we denote, e(L, ǫ) := {x ∈ R n | ∃y ∈ L, |y − x| ≤ ǫ}, the ǫ-extension of L. For two sets L, T ⊂ R n , we denote d H (L, T ) := inf{ǫ; L ⊂ e(T, ǫ) and T ⊂ e(L, ǫ)}, the Hausdorff distance between L and T . For a family C ⊂ S we say that C is compact if the set is compact with respect to the Hausdorff metric. We say that C is non-degenerate if the relative interior of each simplex s ∈ C is non-empty. For a set C ⊂ S and t > 0, we understand tC as {ts; s ∈ C}.
Our last theorem reads, Theorem 1.3 Let C ⊂ S be a family of simplices. The function is decreasing. Moreover, if C is compact and non-degenerate, then the above function is bounded from above, and thus the limit exists.
Roughly speaking, the above theorem states that one should expect to find a constant number of facets of a given shape at any scale. For example, if n = 3 and the set C consists of all triangles of distance ǫ to an equilateral triangle whose edge has length 1, the theorem suggests that there exists some constant S such that one expects to find approximately one almost-equilateral facet of size between t and tS, for all small enough values of t.
Some of our methods of proof extend a certain formula that appears in [Eld], based on very simple principles from integral geometry. Some of these principles have already been used by Baxter [Bax] in order to study the convex hull of planar random walks. The structure of this note is the following: in section 2, we derive certain estimates for one-dimensional random walks, which will be used later on. In section 3 we establish some formulas concerning the distribution of facets of the polytope K α , which will be one of the central ingredients in our proofs. In section 4,5 and 6 we prove theorems 1.1, 1.2 and 1.3 respectively. Finally, in section 7 we discuss some further possible extensions of our methods and raise some questions for further research.
Throughout this note, the symbols C, C ′ , C ′′ , c, c ′ , c ′′ denote positive universal constants whose values may change between different formulas. Given a subset A ⊂ R n , by conv(A) we denote the convex hull of A, ∂A will denote its boundary, Cl(A) its closure and Int(A), its interior. For a function f : R n → R we write supp(f ) = Cl({x; f (x) = 0}), its support.
Acknowledgements The author wishes to thank Itai Benjamini for fruitful discussions. The author is also very thankful to Christoph Thäle for pointing him to the observation that theorem 1 may be generalized to obtain formulae for all intrinsic volumes.

One dimensional random walks
In this section we derive some estimates concerning one-dimensional random walks.
Let 0 ≤ t 1 ≤ ... ≤ t N ≤ 1 be a Poisson point process on [0, 1] with intensity α and let W (t) be a standard 1-dimensional brownian motion. Consider the random walk W (0), W (t 1 ), ..., W (t N ). By slight abuse of notation, for 1 ≤ j ≤ n, denote W (j) = W (t j ). Let us calculate the probability that W (j) ≥ 0 for all 1 ≤ j ≤ N . Recall the second arcsine law of P.Levi, (see for example [MP], Chapter 5, p. 137). Define a random variable, According to the second arcsine law, X has the same distribution as (1 + C 2 ) −1 where C is a Cauchy random variable with parameter 1. Using the definition of the Poisson distribution, this means that, Now suppose that W (t) is a Brownian bridge such that W (0) = W (1) = 0 and consider the discrete brownian bridge W (0), W (t 1 ), ..., W (t N ), W (1). The cyclic shifting principle (see e.g., [Bax]) is the following observation: for every 0 ≤ s ≤ 1, define Γ s (t) = t+s, where the sum is to be understood as a sum on the torus [0, 1]. Then the function W •Γ s (t)−W (s) has the same distribution as the function W (t). Now, since there is exactly one choice i between 0 and N such that W (t j ) − W (t i ) will be non-negative for every 1 ≤ j ≤ N (where t 0 = 0), it follows that for only one choice of 0 ≤ i ≤ N , the function will be positive for all the points t j − t i , 0 ≤ j ≤ N (where the subtraction is again understood on the torus [0, 1]). Since the points t 1 , ..., t N are independent of the function W (t), it follows that (recall that N was a poisson random variable with expectation α).
We conclude the calculations as a lemma: Lemma 2.1 Let B(·) be a standard brownian motion, let W (·) be a standard brownian bridge on [0, r] and let t 1 , ..., t N be a poisson point process on [0, r] with intensity α. We have, where, Moreover, Next, we will need the following estimate: Lemma 2.2 Let B(·) be a standard brownian motion and let (t 1 , ..., t N ) be a Poisson point process with intensity α on the interval [0, r]. Define, We have, where, and erf is the error function, defined as,

Proof:
We follow lines resembling the ones in [MP,Page 215]. Let u(t, x) be the function satisfying, where V (x) = α1 {x<0} . By the Feynmann-Kac formula (see e.g., [MP,Theorem 7.43,page 214]), one has, By the definition of the poisson process and by the independence of B(t) and the poisson process, Our goal is therefore to estimate u(t, 0). For ρ > 0, we define the Laplace transform of u(·, x). We have, by Leibnitz's theorem (explain why we can assume that u t exists) In other words, The only solution to this ODE which does not grow exponentially with x is, for some A, B ∈ R. The function g(x) should be continuously differentiable at 0, thus, by matching derivatives, we attain which gives, Let γ ∈ R and let F (t) be a function which satisfies, In view of (7), we conclude, The proof is complete.

A formula for the facets
The goal of this section is to derive a formula which will serve as a central ingredient in our theorems.
We begin with some notation. For 0 ≤ s 1 < ... < s n ≤ 1, s = (s 1 , ..., s n ), define F s to be the convex hull of B(s 1 ), ..., B(s n ). This is a.s an (n − 1)-dimensional simplex. Let E(s) be the measure zero event that F s is a facet in the boundary K, and let E α (s) be the event that F s is a facet in the boundary of K α . Let n s be a unit vector normal to F s chosen such that n s , B(s 1 ) ≥ 0, and finally define, For a point s defined as above, we define r(s) = (r 1 , ...r n ) by , which we denote by ∆ n . Analogously, for a point r ∈ ∆ n define by s(r) the corresponding point s = (s 1 , .., s n ). By slight abuse of notation we will also write F r , E(r), E α (r), n r , V (r) and H(r), allowing ourselves to interchange freely between s and r.
We define two random measures: where δ r denotes a Dirac probability measure whose support is the point r. For a Borel subset A ⊂ ∆ n , we define the expected number of facets F r , with r ∈ A, found in the boundary of K α and K respectively.
Denote by W α (r) the measure zero event that the point r ∈ ∆ n is also in the Poisson process corresponding to K α (hence the event that all the points r 1 , r 1 + r 2 , ..., r 1 + ... + r n are in the set Λ α ) and define the measure, Observe that for all α > 0, we have µ α ≪ ν α . Therefore, we may denote Let us try to understand how to calculate p α (r). First note that thanks to an elementary property of a Poisson process, it makes sense to condition on the zero-probability event W α (r). Namely, for a random variable X, we understand the expression E[X| W α (r)] in the following way: define a new point process Λ = Λ ∪ {s 1 (r), ..., s n (r)}, and take the expectation of X, replacing Λ withΛ.
By Fubini's theorem and by the independence of the Poisson points in disjoint intervals, for all α > 0 and for all continuous functions f : ∆ n → R, By taking f (r) = 1, we see that in fact, almost everywhere in ∆ n . Note also that the function f in equation (9) need not be deterministic: it may also depend on B(·) and Λ, as long as for every r ∈ ∆ n , f (r) is measurable.
Our next task is to understand the measure ν α better. To that end, let s = (s 1 , ..., s n ) and ǫ > 0 be such that Then, by the independence of the number of Poisson points on disjoint intervals, By the σ-additivity of ν, it follows that for a measurable A ⊂ Int(∆ n ), where in the last equality we use the fact that the Jacobian of the function r → s(r) is identically one. We learn that, in fact, dν α = α n dr. Therefore, equation (9) becomes, The above formula will play a central role in our proofs. It will serve us to find the expectation of several quantities of interest. For instance, in order to calculate the volume of K, we observe that: (12) Using equation (11), we get In a similar way, we obtain the following formula for the surface area: Next, we would like to derive more explicit expressions for the expectations on the right hand side of the two last formulae. The next lemma follows lines analogous to the ones developed in ( [Eld]): Lemma 3.1 For all r ∈ ∆ n , one has, where Ψ is defined as in lemma 2.1. Moreover, upon conditioning on W α (r), the event E α (r) is independent from F r , and one has

Proof:
Our first goal is to write the event E α (r)|W α (r) as the product of independent events whose probabilities will be calculated using the formulas derived in section 2. The idea which allows us to do this is the following: the representation theorem for the brownian bridge suggests that we may equivalently construct B(t) by first generating the differences B(s j ) − B(s j−1 ) as independent gaussian random vectors, and then "fill in" the gaps between them by generating a brownian motion up to B(s 1 ), a brownian bridge for each 1 < j ≤ n, and a "final" brownian motion between B(s n ) and B(1), all of the above independent from each other. To make it formal, fix r ∈ ∆ n and define s = s(r). For all i, 2 ≤ i ≤ n, we write, By the independence of the differences of a brownian motion on disjoint intervals and by the representation theorem for the brownian bridge, it follows that the variables are all independent, each C i being a brownian bridge and B 0 and B f being brownian motions.
Observe that for all 2 ≤ i ≤ n,C i is a one-dimensional brownian bridge fixed to be zero at its endpoints, and B 0 and B f are one dimensional brownian motions starting from the origin.
A moment of reflection reveals that the event E α (s) is reduced to the intersection of the following conditions, for each possible direction of n s with respect to F s , The functionB 0 is non-negative at all points t j such that t j < s 1 . (iv) The functionB f is non-negative at all points t j such that s n < t j ≤ 1.
As explained above, {C i } n i=2 ,B 0 andB f are independent, thus we can esti-mate p(r) using equations (4) and (6). We get, Note that the factor 2 stems from that fact that n s has two possible directions.
Next, we note that H r = B 0 (0). DefiningẼ α (r) as the event that (iii) above holds, we use lemma 2.2 in order to learn that Since (i), (ii) and (iv) above are independent from (iii) and from B 0 (0), we get Finally, by the rotational invariance of the model, it follows that n s is independent of B 0 , B f and C i . Since the distribution of the facet F s depends only on the projection of the brownian motion onto the subspace n ⊥ s , we learn that V (r) is independent from the events (i)-(iv) above. In particular, A combination of the last two equations gives (16).

Expected volume and surface area
The purpose of this section is to use the technique developed in the previous section in order to obtain a formula for the expected volume of K. The formula will be derived in the following way: First, we can find a formula for the expected volume of K α by combining formula (13) and lemma 3.1. Then, in order to find E[V ol n (K)], we will establish the fact that the latter may be expressed as a limit of the former, by taking α → ∞. This fact is stated in a corollary below. We begin with some notation. For a convex body L and for ǫ > 0, we denote, e(L, ǫ) := {x ∈ R n | ∃y ∈ L, |y − x| ≤ ǫ}, It is easy to verify that, lim ǫ→0 d(K(ǫ)), K) = 0. Therefore, it is enough to show that almost surely, for every ǫ > 0, there exist α 0 such that for every α > α 0 , one has P(K α ⊃ K(ǫ)) > 1 − ǫ.

Evidently, D(θ)
is an open set that contains θ. Next, define, where B(θ, r) is an open spherical cap of radius r, centered at θ. The fact that D(θ) is open implies that r(θ) > 0. Moreover, one may verify that r(θ) is continuous with respect to θ, and therefore attains a minimum, r 0 > 0. Now, take θ 1 , ..., θ M to be an r 0 -net of the sphere. Suppose a set of points x 1 , ..., x M ∈ K satisfy x i ∈ K(ǫ/2, θ i ), and denote C = conv(x 1 , .., x M ). Then for all θ ∈ S n−1 , there exists some i such that θ ∈ B(θ i , r 0 ) which implies that It is therefore enough to show that the following event has probability tending to 1: . By the continuity of B, this set has a positive measure, which means that the probability that one of the points of the Poisson process is in T i tends to 1 as α → ∞. By applying a union bound, it follows that lim α→∞ P (E) = 1, and the lemma is proven.
As a direct corollary, we obtain In view of the above corollary, the proof of theorem 1.1 is reduced to calculating lim α→∞ E[V ol n (K α )]. Recall formulae (13) and (16). The only ingredient we still need is E[V (r)]. The next lemma is a simple calculation.
Lemma 4.2 Let r = (r 1 , ..., r n ) ∈ ∆ n . We have, X where X is a random variable whose distribution does not depend on r.
Proof: Define, s = s(r) and Let Γ 1 , .., Γ n−1 be independent standard gaussian random vectors in R n . By the independence of increments of the Brownian motion on disjoint intervals, we have Observe that the dimension of E i is almost surely n − i. Let u 1 , ..., u n be vectors such that {u 1 , ..., u n−i+1 } is an orthonormal basis of E i−1 . One has, Note that in the last equality we use the fact that Γ i is independent of E i−1 and therefore the vectors u 1 , ..., u n−i+1 can be assumed constant. The above is just the first moment of the χ-distribution with (n − i + 1) degrees of freedom, which is equal to √ 2 Γ((n − i + 2)/2) Γ((n − i + 1)/2) = 2 (n−1)/2 Γ((n + 1)/2).
Plugging this into equation (13), we finally get As mentioned above, we aim at calculating E[V ol n (K)] by means of taking α → ∞. To do this, we would like to use the dominated convergence theorem in order to take the limit inside the integral, so we end up with a simpler integrand. Let us inspect each term separately. First note that, Then, we estimate for all t ≥ 0. We learn that the integrand in (20) is positive and smaller than the term 3 n+1 j=2 1 √ rj , whose integral clearly converges, therefore we may use the dominated convergence theorem. For all r ∈ Int(∆ n ), we calculate, We attain, (interchanging between r 1 and r n+1 ) where B n := {x ∈ R n , |x| ≤ 1}, the unit euclidean ball.
We have established theorem 1.1.

The approximating polytope
The goal of this section is to prove theorem (1.2). Since in the previous section, we already calculated E[V ol n (K)], the derivation of the bounds in the theorem is reduced to obtaining estimates on E[V ol n (K α )], which in turn, are reduced to obtaining respective estimates on (20).
We begin with the lower bound. Inspect equation (20). Our first goal will be to show that the term involving the expression e −αr 1 −1 √ πr1α is small. We calculate, using (21), (here, we used the substitution r i = x 2 i ). It is easy to verify that, 12 √ n √ α ξ n V ol n (B n ) n .
Our next task is to estimate the remaining term. To that end, we note that Φ(x) ≥ erf ( √ x) and use the following well-known estimate: This estimate yields, 1 − e −αrj   dr = (interchanging between r 1 and r n+1 ) Define, An straightforward calculation gives, So we can estimate, Using (24), we attain, Bn e (|x| 2 −1)α ≤ V ol n (B n ) 1 α + 10nα where in the last inequality, we use the legitimate assumption that α is greater than some universal constant. Combining this with (23) yields, Next, we note that the term n j=1 1 − e −αt 2 j is monotone on rays of the form {s(t 1 , ..., t n ); s ≥ 0}, which implies that where Γ = (Γ 1 , ..., Γ n ) is a standard Gaussian random vector. A calculation gives P(|Γ| 2 ≤ 4n) > 1 − e −n . So we get, The above estimate combined with (26) gives, Finally, the last equation along with (20) and (22) give, which implies the lower bound bound.
We continue with the upper bound. Formula (20) and the bound (21) suggest that, (interchanging between r 1 and r n−1 and substituting t 2 Let X = (X 1 , ..., X n ) be uniformly distributed in B n . It is straightforward to verify that We observe that for all k > 1 and for all 0 ≤ t ≤ n/10, one has, It follows that, for all 0 ≤ t ≤ n/10. The last inequality, combined with (27), implies that for all 0 ≤ t ≤ n/10. Using the assumption α < n 3 8 and choosing t = 4 log(n 3 /α) gives The upper bound is established, and we have proven theorem (1.2).

Approximate scaling invariance of the facet distribution
In this section we will prove theorem 1.3.
We begin by fixing some C ⊂ S. Recall the definition of the measure q(·) (equation 8). We have, Our main ingredient will be the following lemma connects between the facets of K α and the those of K: Lemma 6.1 Let f : ∆ n → R be a continuous (random) function such that for all r ∈ ∆ n , f (r) is measurable with respect to the Brownian motion B(·) and such that max r∈∆n |f (r)| ≤ 1 almost surely. We have, In particular, for any Borel set A ⊂ ∆ n , one has We postpone the proof of this lemma to the end of the section.
Using the above lemma with f (r) = 1 {Fr∈C} , we attain, Since the equivalence class of F r and the event E α (r) are independent (by lemma 3.1), it follows that, Let Γ 1 , ..., Γ n−1 be independent standard gaussian random vectors. For a point r ∈ ∆ n , we define so X r has the same distribution as F r , up to a translation. Using formula (15), equation (29) becomes, The bound (21) suggests that the dominated convergence theorem may be used to attain, Recall that for a set C ⊂ S and t > 0, we understand tC as {ts; s ∈ C}. Note that X tr ∼ √ tX r . Therefore, by substituting r i = λw i for 2 ≤ i ≤ n in the last integral, we attain that for all λ ≥ 1, It follows that M K (tC) is a decreasing function of t, which means that, lim t→0 M K (tC) exists in the wide sense. This completes the first part of the theorem.
The second part of the theorem will follow directly from the next technical lemma.
Proof: For a simplex s ∈ C, let M (s) and m(s) denote the length of the longest and shortest two-dimensional edge of s, respectively. Since C is compact we may define, Along with the assumption that the simplices are non-degenerate (e.g. have a non-empty interior), the reader can easily verify that m > 0. Fix (r 1 , ..., r n ) = r ∈ R n + and for all 1 ≤ i ≤ n let t i = i j=1 r i . Let Γ be a standard Gaussian random vector in R n . Recall the definition of X r in equation (30). We estimate, for some constants c 1 , C 1 > 0. On the other hand, we can write for some constants c 2 , C 2 > 0 (which may depend on M and n). Using these two estimates, we obtain, for some constants c, C > 0. We obtain, for some constant C ′ > 0. The above integral obviously converges, and the lemma is proven.
Remark 6.1 The method above may actually be used to find a precise formula for, say, the expected number of facets of K whose volume is between two given constants, in terms of the distribution function of the product of χ-random variables. As another example, by taking f (r) = V (r) 2 in lemma 6.1, another quantity which we can easily calculate using this method is the expected volume of the facet containing a point x ∈ ∂(K) when x is uniformly generated on the set ∂ (K). Quantities of this sort may serve us to attain a little more information on the distribution of facets of K.
Proof of lemma 6.1: We divide the proof into several steps.
Step 1. We start with showing that for all A ⊂ Int(∆ n ) compact, one has Assume that q(A) = 0. For any r ∈ A, denote, The assumption q(A) = 0 implies that T (r) = ∅ for all r ∈ A. By the continuity of the Brownian motion, it follows that T (r) is an open set and moreover λ(T (r)) (namely, the Lebesgue measure of T (r)) is a positive and continuous function and so attains a minimum on A, which we denote by ǫ. Now, for every r ∈ A, letT (r) be some closed subset of T (r) whose Lebesgue measure is at least ǫ/2. It is easy to check that for all r ∈ ∆ n there exists an open neighborhood, N (r) such that r ′ ∈ N (r) ⇒T (r) ⊂ T (r ′ ).
Since A is compact, there exists a finite set r 1 , ..., r N ∈ A such that Now, suppose that for each 1 ≤ i ≤ N , one has Λ α ∩T (r i ) = ∅. Then for all r ∈ A we have T (r) ⊃T (r j ) for some 1 ≤ j ≤ N , which implies that T (r) ∩ Λ α = ∅, which in turn means that E r (α) does not hold. It follows that, Since Λ α is increasing with α, it follows that the events {Λ α ∩T i = ∅} are increasing with α, and we get lim sup α→∞ µ α (A) = 0.
Step 2 We show that if A ⊂ Int(∆ n ) is compact and f is positive, continuous and supported in A then, To prove this, we observe that for almost every point r ∈ ∆, there exists a neighborhood M (r) such that Using the result of [Eld] we learn that almost surely, for α large enough one has 0 ∈ K α . Therefore, by definition of M (r) one has lim sup α→∞ q α (M (r)) ≤ 1.
Denote S = supp(q) ∩ A. For all ǫ > 0, define where B(r, ǫ) denotes the open euclidean ball of radius ǫ centered at r. Then due to (33), we learn that Next, since M is open by definition, using (31), we have lim sup α→∞ A\Mǫ f (r)dq α (r) = 0.
Evidently, the set function lim sup α→∞ q α (·) is sub-additive. Thus, the last two equations imply that for all ǫ > 0, By using the continuity of f and taking ǫ → 0 we attain (32).
Step 3: We show that if f is a continuous then, We may clearly assume that f (r) ≤ 1 for all r ∈ ∆ n . Assume by contradiction that one has Recall that V (r) = V ol n−1 (F r Define L = {r ∈ ∆ n | V (r) > 0}. For now, we assume that supp(f ) ⊂ L. We denote m = min r∈supp(f ) V (r). Since supp(f ) is compact (by definition, it is a closed set), we know that m > 0. Write h(r) = V (r) − m 2 f (r) and note that h(r) ≥ 0 for all r ∈ ∆ n . Equations (36) and (35) This contradicts (32). Therefore, we have shown (34) for all functions f supported in L. But since L is an open set since the measure of ∆ n ∩ L is zero, we learn that it is true for all functions.
Step 4 To finish the proof of the lemma, suppose that f is supported in some compact A ⊂ Int(∆ n ). We argue that the dominated convergence theorem may be used to show that, Indeed, using the assumption that |f (r)| ≤ 1 for all r ∈ ∆ n , we attain As suggested by equations (20) and (21), there exists a constant C > 0 such that for all α > 0 Consequently, the family { ∆n f (r)dq α (r)} α may be bounded from above by an integrable expression, and (39) follows. We now combine (39), (34) and (11) to get By the σ-additivity of both sides of the equation with respect to f , we conclude that the equality holds for any bounded continuous function f . The lemma is complete.
We finish this section with a small remark. As a consequence of the above lemma, and by corollary 4.1, we have Note that a-priori, the quantity ∆n V (r)dq(r) need not be equal to the surface area of K, because some of its surface area may be contained in facets of dimension smaller than n − 1. We therefore have the following corollary to the above lemma: Corollary 6.1 Let A ⊂ ∂K be the set of points not contained in the interior of any (n − 1)-dimensional facet of K. We have, V ol n−1 (A) = 0.

Higher moments
In this paper, we derived formulas for the expectation (i.e, the first moment) of the volume and surface area of K. It may be interesting to investigate the behaviour of higher moments. In particular, it may be interesting to ask, for example, how concentrated the volume of K is around its mean.
One possible strategy of finding the second moment is by using an analogous formula to (20). It can be seen that, in the notations of section 3, It seems rather hard to find a precise formula for the integrand. However, when the dimension n is large, it is possible to show using standard techniques related to high-dimensional measure concentration that for a typical choice of x, y ∈ ∆ n × ∆ n (i.e, with high probability), the facets F x and F y will be approximately orthogonal to each other. Now, note that the random variables R(x) and 1 Eα(x) depend only on B(t), n x while the variables R(y) and 1 Eα(y) depend only on B(t), n y . It follows that when n x ⊥ n y , these two pairs are mutually independent. Since n x and n y are almost orthogonal, it is reasonable to suspect that for a typical pair x, y one would have In some sense, the above would imply that E[V ol n (K)] 2 is close to E[V ol n (K) 2 ] when the dimension is large. This suggests that the answer to the following question may be positive:

Smoothness
In 1983, El Bachir ( [ElB]) proved the assertion of P. Lévi that almost surely, the convex hull of a planar brownian motion has a smooth boundary. Later, in 1989, Cranston, Hsu and March ([CHM]) showed that in fact, it is exactly Hölder 1 1 2 -smooth. A natural question would be about the extension of these facts to higher dimensions: Question 7.2 Does the body K have a smooth boundary for any dimension?
It doesn't seem straightforward to adapt their methods even to the three dimensional case: their proofs rely on the fact that if a 2-dimensional convex hull has a "corner", then the directions of its supporting hyperplanes will contain some interval, which will in turn contain a rational direction. This fact allows them to express the smoothness of the boundary as a countable intersection of events, each depending on a single direction. It is easy to see that in 3-dimensional space, this is already not the case: it is not hard to construct a body whose boundary is not smooth, but a uniformly generated random 2dimensional projection of this body will almost surely be smooth.
The following heuristic argument may suggest that the boundary is, in fact, smooth in higher dimensions: For two (n − 1)-dimensional facets s, t of a polytope P , we say that s, t are neighbors if the intersection of s, t has Hausdorff dimension n − 2. The first step is to try to prove that for any ǫ > 0, the probability that there exists at least one pair of n − 1-dimensional facets of K α such that the angle between the two is at least ǫ goes to zero as α → ∞. The idea is the following: in the notation of section 3, note that any choice of two neighboring facets of K α corresponds to a choise of n + 1 points from the process Λ α . In other words, it corresponds to the choice of two point r, s ∈ ∆ n such that t(r) and t(s) differ by only one coordinate.
Consider the event E that both F r and F s are facets of K α and that the angle between the two is more than ǫ. By the representation theorem for the brownian bridge, it follows that one can first generate the points B(t) for t ∈ t(r) ∪ t(s), and then "fill in" the missing gaps by brownian bridges, as carried out in the proof of lemma 3.1. Now, project the brownian motion onto the two dimensional subspace spanned n r and n s . Following the same lines as the proof of lemma 3.1, we see that the event E is reduced to the event that n − 1 independent discrete brownian bridges and two brownian motions all stay in a wedge of angle π − ǫ (in lemma 3.1, the event E α (r) was equivalent to the fact that they stay in a half-space).
The next step would be to generalize the bounds in lemma 2.1 to a wedge rather than a half-space. Considering the bounds obtained in [D] as well as by the conformal invariance of the brownian motion, it is reasonable to expect that the probability of an α-step random walk to stay in a wedge of angle θ is of the order α − π 2θ , and that for an α-step discrete brownian bridge to stay in such a wedge, would be of order α − π θ . If these estimates are indeed correct, plugging them in the analogue of equation (15) for wedges should give roughly, If the above bound is true, it would imply that, as α → ∞. This means that when α is large, one should expect K α to be "smooth" in the sense that any neighboring two faces have angle less than ǫ between them. Next, an analogue of lemma 6.1 may be used to show that this property is preserved when passing to the limit: Indeed, the lemma shows that any facet of K already becomes a facet of K α for α large enough. Therefore, if K has two neighboring facets with some angle ǫ > 0 between them, then K α will have two such facets for all α large enough, and we would arrive at a contradiction.

Neighborliness of the approximating polytope
A polytope P ⊂ R n is said to be k-neighborly if for any choice of k vertices of P , v 1 , ..., v k , the simplex conv(v 1 , ..., v k ) is a facet of P . The concept of neighborlyness is related to the ability of linear programming to find solutions to systems of underdetermined linear equations (see [DT]).
We believe that a slight generalization of the method introduced in [Eld, section 2] may be used to show that when α is a polynomial of n, the polytope K α is (cn/ log 2 n)-neighborly, with probability approaching 1 as n → ∞. The idea of proof is as follows: Fix some k < n and take α = n 10 , say. Let 0 ≤ s 1 < ... < s k ≤ 1 be a selection of k points from Λ α . Define F = conv(B(s 1 ), ..., B(s k )). Let us try to understand the event that F is contained in the boundary of K α . By the representation theorem of the Brownian bridge, one may first generate the points B(s 1 ), ..., B(s k ) and then "fill in" the gaps with brownian bridges, as carried out in the proof of lemma 3.1. In view of this, and by considering the projection of the Brownian motion onto F ⊥ , the event above is reduced to the fact that k − 1 discrete brownian bridges of length smaller than n 10 in R n−k and two random walks are all contained in some open halfspace of R n−k . At this point, theorem 2.1 in [Eld] comes to our service. According to this theorem, for a random walk of polynomial length in R n with probability 1 − n −10 , there exists a unit vector whose scalar product with any internal point of the random walk is of order log n. By proving an analogous result for a discrete brownian bridge, one would be able to combine log n such vectors together to create a vector separating these bridges from the origin, thus proving that F is in the boundary of K α .
In some sense, the property of a polytope being k-neighborly is contradictory to the fact that it approximates a smooth convex body: it is not hard to realize that as a family of polytopes approaches a smooth convex body, they become less neighborly in some sense. Therefore, the above fact may be interesting considering the fact that K n 4 is already a good volumetric approximation for the polytope K which is obviously not 2-neighborly (and may, in fact, be smooth).

Comparison with a Gaussian Polytope
Fix a dimension n. For an integer α ∈ N, let Γ 1 , ..., Γ α be independent standard Gaussian random vectors in R n . We define G α = conv(Γ 1 , ..., Γ α ), the convex hull of independent Gaussian points. The object G α is usually referred to as a Gaussian polytope. The study of the Gaussian polytope began in the 60's by Rényi and Sulanke, and ever since it has been deeply investigated (see [CMR] for a survey).
For convex geometers, a central motivation in the investigation of random polytopes stems from the fact that random objects often admit a rather pathological behavior and thus often serve as counterexamples to certain conjectures. A-priori, one may have expected that the Gaussian polytope may be used to serve as a counter example for certain phenomena related to the distribution of mass on convex bodies. Alas, results such as [KK] and [F] suggest that this object admits a quite regular and symmetric nature. It may therefore be interesting to try to find a construction of random polytopes that are, in some sense, as irregular and asymmetric as possible. Some of the estimate we obtained here may point to the fact that the behavior of the polytope K α is less regular than the behavior of G α . We list some possible qualitative differences between those two constructions.
It can be seen (see e.g., [F], theorem 3) that the facets of G n admit a rather regular behavior in the sense that for any given family of simplices C, the function M Gα (tC) is rather concentrated around a specific value of t. In other words, there is a typical "correct" scale at which one expects to find most of the facets of G α . On the other hand, by theorem 1.3, we know that when α is rather large, one expects to find facets of the same shape at a wide range of scales, which suggests a less regular behavior.
As shown in [KK], the covariance matrix of a uniform point randomly generated of G α is not far from a multiple of the identity matrix (for α large enough). This property is sometimes referred to as isotropicity. It can be also seen by [F,Theorem 3] that the covariance matrix of a typical facet of G α is rather isotropic. On the other hand, in view of formula (17), it is reasonable to guess that the covariance matrix of a uniform point on K α , as well as on one of the large-scale facets of K α will have a covariance matrix close to the one of B(·), which is far from isotropic. Note also that the small-scale facets which, in the notations of section 3, have r i ≪ 1 for 2 ≤ i ≤ n, seem to "ignore" the covariance of B(·) and become nearly isotropic again. This also suggests that unlike the gaussian case, if K α is renormalized to be isotropic, the shapes of its smaller facets will be highly correlated.
Both constructions, G α and K α seem to tend to a rather smooth shape as α → ∞: it is well known that the shape of G α becomes quite close to a euclidean ball when the value of α is large enough, namely exponential in n, and K α approaches K, which as the last section suggests, may have a smooth boundary. It is known that G α is a highly-neighborly polytope when n is a proportion of α (see [DT]), but it becomes almost non-neighborly as it approaches a euclidean ball. On the other hand, K n 10 which is, by theorem 1.2, already close in expectation to its "smooth limit", may be a highly neighborly polytope, as the previous subsection suggests.