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Zapiski Nauchnykh Seminarov POMI, 2022, Volume 515, Pages 121–140 (Mi znsl7258)  

Intrinsic volumes of ellipsoids

A. Gusakovaa, E. Spodarevb, D. Zaporozhetsc

a Institute of Mathematical Stochastics, Münster University, Orléans-Ring 10, 48149 Münster, Germany
b Institute of Stochastics, Ulm University, Helmholtzstr 18, 89069 Ulm, Germany
c St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg, Russia
References:
Abstract: We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that
\begin{align*} V_k({\mathcal E})&=\kappa_k\sum\limits_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2) \\&\times\int\limits_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod\limits_{j=1}^d\sqrt{a_j^2t^2+1}} \rm{d}t \end{align*}
for all $k=1,\ldots,d$, where $s_{k-1}$ is the $(k-1)$-th elementary symmetric polynomial and $\kappa_k$ is the volume of the $k$-dimensional unit ball. Some examples of the intrinsic volumes $V_k$ with low and high $k$ are given where our formulae look particularly simple. As an application we derive new formulae for the expected $k$-dimensional volume of random $k$-simplex in an ellipsoid and random Gaussian $k$-simplex.
Key words and phrases: Convex body, intrinsic volume, mixed volume, quermaßintegral, Minkowski functional, support function, mixed discriminant, ellipsoid, polar ellipsoid, hypergeometric $R$-function, Weinstein-Aronszajn identity.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1620
Deutsche Forschungsgemeinschaft EXC 2044–390685587
Foundation for the Development of Theoretical Physics and Mathematics BASIS
Russian Foundation for Basic Research 20-51-12004
The work of ES was supported by EIMI (Leonard Euler International Mathematical Institute, Saint Petersburg), grant No. 075-15-2019-1620 as of 08/11/2019. AG was supported by the DFG under Germany's Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure. The work of DZ was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” and by RFBR and DFG according to the research project No. 20-51-12004.
Received: 01.11.2022
Document Type: Article
UDC: 519
Language: English
Citation: A. Gusakova, E. Spodarev, D. Zaporozhets, “Intrinsic volumes of ellipsoids”, Probability and statistics. Part 33, Zap. Nauchn. Sem. POMI, 515, POMI, St. Petersburg, 2022, 121–140
Citation in format AMSBIB
\Bibitem{GusSpoZap22}
\by A.~Gusakova, E.~Spodarev, D.~Zaporozhets
\paper Intrinsic volumes of ellipsoids
\inbook Probability and statistics. Part~33
\serial Zap. Nauchn. Sem. POMI
\yr 2022
\vol 515
\pages 121--140
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl7258}
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  • https://www.mathnet.ru/eng/znsl/v515/p121
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