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arXiv · 1409.8105

The volume of random polytopes circumscribed around a convex body

Ferenc Fodor·Daniel Hug·Ines Ziebarth
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Abstract

Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We achieve these results by first proving similar statements for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then by polarizing these results.

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Ferenc Fodor, Daniel Hug, Ines Ziebarth. 2014-09-29. The volume of random polytopes circumscribed around a convex body. https://doi.org/10.1112/s0025579315000170

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