Geodesic
Random section and random simplex inequality
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 31, Tome 505 (2021), pp. 162-171 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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. 
@article{ZNSL_2021_505_a9,
     author = {A. E. Litvak and D. N. Zaporozhets},
     title = {Random section and random simplex inequality},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {162--171},
     year = {2021},
     volume = {505},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_505_a9/}
}
TY  - JOUR
AU  - A. E. Litvak
AU  - D. N. Zaporozhets
TI  - Random section and random simplex inequality
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2021
SP  - 162
EP  - 171
VL  - 505
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2021_505_a9/
LA  - ru
ID  - ZNSL_2021_505_a9
ER  - 
%0 Journal Article
%A A. E. Litvak
%A D. N. Zaporozhets
%T Random section and random simplex inequality
%J Zapiski Nauchnykh Seminarov POMI
%D 2021
%P 162-171
%V 505
%U http://geodesic.mathdoc.fr/item/ZNSL_2021_505_a9/
%G ru
%F ZNSL_2021_505_a9
A. E. Litvak; D. N. Zaporozhets. Random section and random simplex inequality. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 31, Tome 505 (2021), pp. 162-171. http://geodesic.mathdoc.fr/item/ZNSL_2021_505_a9/

[1] H. Busemann, “Volume in terms of concurrent cross-sections”, Pacific J. Math., 3 (1953), 1–12 | DOI | MR | Zbl

[2] H. Busemann, E. Straus, “Area and normality”, Pacific J.Math., 10 (1960), 35–72 | DOI | MR | Zbl

[3] E. Grinberg, “Isoperimetric inequalities and identities fork-dimensional cross-sections of convex bodies”, Math. Ann., 291 (1991), 75–86 | DOI | MR

[4] H. Furstenberg, I. Tzkoni, “Spherical functions and integralgeometry”, Israel J. Math., 10 (1971), 327–338 | DOI | MR | Zbl

[5] R. Schneider, “Inequalities for random flats meeting a convex body”, J. Appl. Probab., 22 (1985), 710–716 | DOI | MR | Zbl

[6] M. Crofton, “Probability”, Encyclopaedia Brittanica, 19, 1985, 758–788

[7] H. Hadwiger, “Ueber zwei quadratische Distanzintegrale für Eikörper”, Arch. Math. (Basel), 3 (1952), 142–144 | DOI | MR | Zbl

[8] G. Chakerian, “Inequalities for the difference body of a convexbody”, Proc. Amer. Math. Soc., 18 (1967), 879–884 | DOI | MR | Zbl

[9] J. Kingman, “Random secants of a convex body”, J. Appl. Prob., 6 (1969), 660–672 | DOI | Zbl

[10] R. Gardner, “The dual Brunn-Minkowski theory for boundedBorel sets: dual affine quermassintegrals and inequalities”, Adv. Math., 216 (2007), 358–386 | DOI | MR | Zbl

[11] S. Dann, G. Paouris, P. Pivovarov, “Bounding marginaldensities via affine isoperimetry”, Proc. Lond. Math. Soc., 113 (2016), 140–162 | DOI | MR | Zbl

[12] R. Schneider, W. Weil, Stochastic and integral geometry, Springer-Verlag, Berlin, 2008 | Zbl

[13] H. Groemer, “On some mean values associated with a randomlyselected simplex in a convex set”, Pacific J. Math., 45 (1973), 525–533 | DOI | MR | Zbl

[14] F. Götze, A. Gusakova, D. Zaporozhets, “Random affine simplexes”, J. Appl. Probab., 56 (2019), 39–51 | DOI | MR | Zbl

[15] H. Busemann, The geometry of geodesics, Academic Press Inc., New York, 1995