On the deterministic interior body of random polytopes

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 $K_N$.