Volume radius of a random polytope in a convex body

Abstract

Let $K$ be a convex body in ${\bb R}^n$ with volume $|K| = 1$. We choose $N$ points $x_1,\ldots, x_N$ independently and uniformly from $K$, and write $C(x_1,\ldots, x_N)$ for their convex hull. If $n(\log n)^2 \leqslant N \leqslant \exp(c_1n)$, we show that the expected volume radius \[ {\bb E}_{\frac{1}{n}}(K,N)=\int_K\cdots\int_K |C(x_1,\ldots, x_N)|^{\frac{1}{n}}dx_N\cdots dx_1 \] of this random $N$-tope can be estimated by \[ c_1\frac{\sqrt{\log(N/n)}}{\sqrt{n}}\leqslant{\bb E}_{1/n}(K,N)\leqslant c_2L_K\frac{\log(N/n)}{\sqrt{n}}, \] where $c_1, c_2$ are absolute positive constants and $L_K$ is the isotropic constant of $K$.