On the deterministic interior body of random polytopes

TL;DR

This paper investigates the asymptotic shape of random polytopes formed by convex hulls of i.i.d. vectors, establishing probabilistic inclusions related to measure depth without extra assumptions.

Contribution

It generalizes previous results by providing bounds on the interior of random polytopes for arbitrary measures, using depth functions and comparing them to other convex bodies.

Findings

For N ≥ c(β)n, the random polytope contains a depth-based convex set with high probability.

The approach does not require additional assumptions on the measure μ.

Results improve and extend previous work on the shape of random polytopes.

Abstract

Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of independent copies of a random vector $X$ in $\mathbb{R}^n$. We revisit the question to determine the asymptotic shape of the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$ where $N>n$. We show that for any $\beta\in (0,1)$ there exists a constant $c(\beta)>0$ such that the following holds true: If $\mu $ is a Borel probability measure on ${\mathbb R}^n$ then, for all $N\geq c(\beta)n$ we have that $K_N\supseteq T_{\beta\ln(\frac{N}{n})}(\mu)$ with probability greater than $1-\exp(-\tfrac{1}{2}N^{1-\beta}n^{\beta})$, where $T_p(\mu)$ is the convex set of all points $x\in\mathbb{R}^n$ with half-space depth greater than or equal to $e^{-p}$. Our approach does not require any additional assumptions about the measure $\mu$ and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family $\{T_p(\mu)\}_{p>0}$ to other natural families of convex bodies associated with $\mu$, such as the $L_p$-centroid bodies of $\mu$ or the level sets of the Cram\'{e}r transform of $\mu$, and use this information in order to estimate the size of a random $K_N$.