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arXiv · 0906.0309

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Imre Bárány·Ferenc Fodor·Viktor Vígh
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Abstract

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of B\'ar\'any and Larman, and the Efron-Stein jackknife inequality.

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Imre Bárány, Ferenc Fodor, Viktor Vígh. 2009-06-01. Intrinsic volumes of inscribed random polytopes in smooth convex bodies. https://doi.org/10.1239/aap/1282924055

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