Phase transition for the volume of high‐dimensional random polytopes

Abstract

The beta polytope P n, is the convex hull of n i.i.d. random points distributed in the unit ball of R according to a density proportional to (1 − ||x|| 2 ) if > −1 (in particular, = 0 corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if = −1. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when = 0, their number of vertices.

Threshold Phenomena for Random Cones

“…Proof of Theorem 1. 4 The first part of Theorem 1.4 follows from Theorem 1.5, which will be proven below. For the second part of the proof, we assume that d/N → δ as d → ∞, with 0 ≤ δ < 1/2.…”

“…random points with either Gaussian distribution or uniform distribution on the unit sphere. In [3,4], the points have a beta or beta-prime distribution. The paper [5] studies facet numbers of convex hulls of random points on the unit sphere in different regimes.…”

On a game of chance in Marc Elsberg’s thriller “GREED”

“…However, the final score is only larger than 100 if k D 2. This event occurs with probability 1 4 implying that the probability of a loss is given by 3 4 . After n D 6 rounds, the expected net profit is negative for the first time.…”

“…Motivated by our observations on the initial scenario described by Elsberg and a first quantitative analysis in Sect. 3, we generalise the underlying parameters of this game of chance (see Sect. 4).…”

Half-space depth of log-concave probability measures

“…A similar result holds true for the expected volume of random polytopes with vertices uniformly distributed in the cube B n ∞ ; the corresponding value of the constant κ is κ = 2π/e γ+1/2 , where γ is Euler's constant. Further sharp thresholds of this type have been given; see [12] for the case where X i have independent identically distributed coordinates supported on a bounded interval, and the articles [18] and [3], [4] for a number of cases where X i have rotationally invariant densities. Exponential in the dimension upper and lower thresholds are obtained in [11] for the case where X i are uniformly distributed in a simplex.…”