Concentration of random polytopes around the expected convex hull

We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a corresponding strong law of large numbers.


Introduction
Let d ∈ N and let µ be a probability measure on R d with a log-concave density f = dµ/dx, i.e. − log f is a convex extended real valued function. Let n ≥ d + 1 and let (X i ) n 1 denote an i.i.d. sequence of random vectors with common distribution µ. The convex hull is a random polytope and, as such, is a random element w.p.1 of the space K d of all convex bodies in R d (compact convex sets with non-empty interior). There are various metrics and metric-like functions on K d , such as the Hausdorff distance d H and the Banach-Mazur distance δ BM (for origin symmetric bodies). We refer the reader to [25] for general background on convex bodies, and to [18] specifically for metric, and other, structures on K d .
It was shown in [12] that if n ≥ c exp(exp(5d)), then with probability at least 1 − 3 d+3 (log n) −1000 , there exists x ∈ R n and λ ≤ 1 + c ′ d 2 log log n log n such that λ −1 (F 1/n − x) + x ⊆ P n ⊆ λ(F 1/n − x) + x where c, c ′ > 0 are universal constants and F 1/n is the floating body defined by where the intersection runs through the collection of all closed half-spaces H of µ-mass at least 1 − δ (δ < e −1 ). The body F 1/n was originally defined by Schütt and Werner [26] in the case of Lebesgue measure on a convex body and has often been used to model random polytopes, see for example [4,5,31].
Being log-concave, the density f decays at least as quickly as an exponential function. Any bound on the decay rate of f translates to a bound on the Hausdorff distance d H (P n , F 1/n ). For example if the tails of µ are sub-Gaussian (with universally bounded constants), then diam(F 1/n ) ≤ c(log n) 1/2 and (2) translates to where c, c ′ > 0 are universal constants. This is an embodiment of the concentration of measure phenomenon: the polytope P n , as a random element of the metric space (K d , d H ), is concentrated around F 1/n . In the case d = 1, P n reduces to the interval and we see that the above mentioned result generalizes a theorem of Gnedenko [14] on concentration of the maximum and minimum of a large i.i.d. sample (under rapid decay of the tails of µ). Other multivariate analogs of Gnedenko's law of large numbers are included in [13] for the multivariate normal distribution, [17] for Gaussian measures on infinite dimensional spaces, [8,10,11] for regularly varying distributions, and [19,22] for more general distributions. The proof of (2) was complicated by the fact that there is no convenient expression for the support function of the floating body, In this paper we study concentration of P n around the expected convex hull which is easily seen to be a convex body with support function Using the expected convex hull leads to a streamlined proof of (2). The notion of the expectation of a random convex body follows the theory of integrals of set valued functions, see for example [1,3,9,20] and the references therein. It was used in [2] for the purpose of a Kolmogorov strong law of large numbers and has appeared as an approximant to floating bodies in bounded domains [6], as well as in other contexts e.g. [15,16,27,28,29,30,32].
In the original paper [12] we were mainly interested in a quantitative dependence on n. Although our bounds included dependence on dimension, the required sample size was very large. Theorem 1 includes improved bounds on the required sample size and is more in the spirit of the high dimensional theory. The quantitative dependence that we achieve is essentially the same as that in Dvoretzky's theorem, see for example [24]. This result should also be compared to the main result in [7].
To make the present exposition brief, we refer the reader to [12] for a more detailed discussion.

Main results
Theorem 1 Let d ∈ N and let µ be a log-concave probability measure on R d with center of mass at the origin and non-singular covariance matrix. Consider any ε ∈ (0, 1/2) and let n ≥ exp(7dε −1 log ε −1 ). Let (X i ) n 1 be an i.i.d. sample from µ, P n = conv{X i } n 1 , and let EP n denote the expected convex hull as defined by (5). With probability at least 1−3n −ε/4 , The following Corollary, which is similar to the main result in [6], is a consequence of Lemma 4.
Corollary 2 Let d ∈ N and let µ be a log-concave probability measure on R d with center of mass at the origin and non-singular covariance matrix. Let EP n denote the expected convex hull as defined by (5), and let F 1/n denote the floating body defined by (3). Then provided n ≥ 12, (1 − 3/ log n)EP n ⊆ F 1/n ⊆ (1 + 1/ log n)EP n Theorem 3 Let d ∈ N and let µ be a log-concave probability measure on R d with center of mass at the origin and non-singular covariance matrix. Let (X i ) ∞ 1 be an i.i.d. sample from µ, and let (P n ) ∞ n=d+1 and (EP n ) ∞ 3 be the random polytopes and expected convex hulls defined by (1) and (5) respectively. Then with probability 1, there exists N ∈ N such that for all n ≥ N,

Notation
If J is the cumulative distribution function associated to a probability measure µ on R, then the generalized inverse J −1 : (0, 1) → R is defined as If µ has a log-concave density function then J(J −1 (t)) = t for all t ∈ (0, 1) and J −1 (J(x)) = x for all x in the support of µ. If (Y i ) n 1 is an i.i.d. sample from µ, then Y (n) = max 1≤i≤n Y i denotes the n th order statistic.
If K ⊂ R d is a convex body then the function is known as the support function of K. If 0 ∈ int(K) then the Minkowski functional is defined as x K = min{λ ≥ 1 : x ∈ λK} and the support function is the Minkowski functional of the polar body i.e. h K (·) = · K • . In the case when K is centrally symmetric, i.e. K = −K, then h K (·) and · K are norms.

Proofs
The following lemma is a natural extension of Lemma 7 in [12].
Lemma 4 Let µ be a probability measure on R with mean 0 and log-concave density f = dµ/dx. Let n ≥ 12 and let (Y i ) n 1 be an i.i.d. sample from µ. Then for all t > 0, Proof. Let J be the common distribution function of each Y i . Let f n and J n denote the density and distribution function of Y (n) , Since f is log-concave, so is J (see for example Theorem 5.1 in [21] or Lemma 5 in [12]). The product of log-concave functions is certainly log-concave, and therefore so is f n . By a standard result, see for example Lemma 5.4 in [21], . Just as the left tail J is log-concave, so is the right tail 1 − J, and the function u(t) = − log(1 − J(t)) is convex. This implies that, (7) follows. Again by convexity of u, which translates to Now, As before, J n (J −1 (1 − 9/(20n))) = (1 − 9/(20n)) n > 1 − e −1 , so EY (n) ≤ J −1 n (1 − e −1 ) < J −1 (1 − 9/(20n)) and (8) (7) of Lemma 4 implies that J −1 (1−1/n) ≤ (1+1/ log n)Eγ (n) . The result now follows from the definitions of F 1/n and EP n , see (3) and (5).
The following lemma appears as Lemmas 4.10 and 4.11 in [23] under the assumption that K is centrally symmetric. We sketch the proof to show that it can also be used in the non-symmetric case.
Lemma 5 Let K ⊂ R d be any convex body with 0 ∈ int(K) and 0 < ε < 1/2. Then there exists a set N ⊂ ∂K with |N | ≤ (3/ε) d such that for all θ ∈ ∂K there exist sequences Proof. Consider a subset N ⊂ ∂K, minimal with respect to set inclusion, with the following property: for all z ∈ ∂K there exists ω ∈ N such that z − ω K ≤ ε. Such a set can easily be constructed recursively, and we shall refer to N as an ε-net. Note that since K may be non-symmetric, we may have z − ω K = ω − z K and order becomes important. By the standard volumetric argument |N | ≤ (3/ε) d . By the defining property of N , for all x ∈ R d there exists ω ∈ N such that Now consider θ ∈ ∂K. By (9) there exists ω 0 ∈ N such that θ − ω 0 K ≤ ε. By applying (9) again, there exists Iterating this procedure defines a sequence (ω i ) ∞ 0 such that for all N ∈ N, Proof of Theorem 1. Set δ = 3n −ε/(4d) and let N ⊂ ∂((EP n ) • ) be a δ-net as in Lemma 5. By the bounds imposed on n, δ ≤ ε/5 < 1/10. From the union bound and Lemma 4, the following event occurs with probability at For any θ ∈ ∂((EP n ) • ), write θ = ω 0 + ∞ 1 δ i ω i , with ω i ∈ N and 0 ≤ δ i ≤ δ i for all i. By the triangle inequality and (10), and the result follows. Proof of Theorem 3. Here d and µ are fixed, and we treat n → ∞ as a variable. From comparing successive terms in the binomial theorem and using the fact that n −k n k is a decreasing function of k, for all δ ∈ (0, 1/2) 1 n(log n) 2 < ∞ Therefore, by the Borel-Cantelli lemma, with probability 1 there exists N (1) ∈ N such that for all n ≥ N (1) , P n ⊆ (1 + 8(log log n)/ log n)EP n For each n ∈ N, let E n be the event that (11) holds. Consider any sufficiently large (deterministic) n ∈ N. Set ε = 3(log log n)/ log n and δ = 3 exp(−n −ε/2 /(6d)). Let N ⊂ ∂((EP n ) • ) be a δ-net as in Lemma 5. As before, δ ≤ ε/10 ≤ 1/20. By the union bound and Lemma 4, the following event, to be denoted F n , occurs with probability at least 1 − (3/δ) d exp(−n ε/2 /3) ≥ 1 − exp(−n ε/2 /6) ≥ 1 − n −2 : for all ω ∈ N , The Borel-Cantelli lemma again implies that with probability 1 there exists N (2) ∈ N such F n occurs for all n ≥ N (2) . For all n ≥ max{N (1) , N (2) }, E n ∩ F n occurs, and expressing an arbitrary θ ∈ ∂((EP n ) • ) as θ = ω 0 + ∞ 1 δ i ω i as in Lemma 5 and using the triangle inequality, (6).