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Zapiski Nauchnykh Seminarov POMI, 2021, Volume 505, Pages 162–171
(Mi znsl7129)
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Random section and random simplex inequality
A. E. Litvakab, D. N. Zaporozhetsc a Dept. of Math. and Stat. Sciences, University of Alberta, Edmonton, AB, Canada, T6G 2G1
b Saint Petersburg State University
c St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Abstract:
Consider some convex body $K\subset\mathbb R^d$. Let $X_1,\dots, X_k$, where $k\leq d$, be random points independently and uniformly chosen in $K$, and let $\xi_k$ be a uniformly distributed random linear $k$-plane. We show that for $p\geq -d+k+1$,
$$
\mathbf E |K\cap\xi_k|^{d+p}\leq c_{d,k,p}\cdot|K|^k \mathbf E |\mathrm{conv}(0,X_1,\dots,X_k)|^p,
$$
where $|\cdot|$ and $\mathrm{conv}$ denote the volume of correspondent dimension and the convex hull. The constant $c_{d,k,p}$ is such that for $k>1$ the equality holds if and only if $K$ is an ellipsoid centered at the origin, and for $k=1$ the inequality turns to equality.
If $p=0$, then the inequality reduces to the Busemann intersection inequality, and if $k=d$ – to the Busemann random simplex inequality.
We also present an affine version if this inequality which similarly generalizes the Schneider inequality and the Blaschke-Grömer inequality.
Key words and phrases:
Blaschke-Grömer inequality, Blaschke-Petkantschin formula, Busemann intersection inequality, Busemann random simplex inequality, convex hull, expected volume, Furstenberg-Tzkoni formula, random section, Schneider inequality.
Received: 11.11.2021
Citation:
A. E. Litvak, D. N. Zaporozhets, “Random section and random simplex inequality”, Probability and statistics. Part 31, Zap. Nauchn. Sem. POMI, 505, POMI, St. Petersburg, 2021, 162–171
Linking options:
https://www.mathnet.ru/eng/znsl7129 https://www.mathnet.ru/eng/znsl/v505/p162
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Abstract page: | 129 | Full-text PDF : | 33 | References: | 18 |
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