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Volume radius of a random polytope in a convex body

Published online by Cambridge University Press:  10 March 2003

A. GIANNOPOULOS
Affiliation:
Department of Mathematics, University of Crete, Iraklion, Greece. e-mail: apostolo@math.uch.gr
A. TSOLOMITIS
Affiliation:
Department of Mathematics, University of the Aegean, Samos, Greece. e-mail: atsol@iris.math.aegean.gr

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\[c_1\frac{\sqrt{\log(N/n)}}{\sqrt{n}}\leqslant{\bb E}_{1/n}(K,N)\leqslantc_2L_K\frac{\log(N/n)}{\sqrt{n}}, \]where $c_1, c_2$ are absolute positive constants and $L_K$ is the isotropic constant of $K$.

Information

Type
Research Article
Copyright
2003 Cambridge Philosophical Society

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