Let K ⊂ \({\mathbb{R}}^{d}\) be a convex body. Let X1, . . . , Xk, where k ≤ d, be random points independently and uniformly chosen in K, and let ξk be a uniformly distributed random linear k-plane. It is proved that if p ≥ −d + k + 1, then
\({\mathbb{E}}{\left|K\cap {\xi }_{k}\right|}^{d+p}\le {c}_{d,k,p}\cdot {\left|K\right|}^{k}{\mathbb{E}}{\left|{\text{conv}}\left(0,{X}_{1},\dots ,{X}_{k}\right)\right|}^{p},\)
where | · | and conv denote the volume of the corresponding dimension and the convex hull, respectively. The constant cd,k,p is such that if k > 1, then the equality holds if and only if K is an ellipsoid centered at the origin, and if k = 1, then the inequality turns to equality.
The inequality reduces to the Busemann intersection inequality if p = 0 and to the Busemann random simplex inequality if k = d.
An affine version of this inequality which in a similar way generalizes the Schneider inequality and the Blaschke-Grömer inequality is also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Busemann, “Volume in terms of concurrent cross-sections,” Pacific J. Math., 3, 1–12 (1953).
H. Busemann, The Geometry of Geodesics, Academic Press Inc., New York, N. Y. (1955).
H. Busemann and E. G. Straus, “Area and normality,” Pacific J. Math., 10, 35–72 (1960).
G. D. Chakerian, “Inequalities for the difference body of a convex body,” Proc. Amer. Math. Soc., 18, 879–884 (1967).
M. Crofton, “Probability,” in: Encyclopaedia Brittanica, 19 (1985), pp. 758–788.
S. Dann, G. Paouris, and P. Pivovarov, “Bounding marginal densities via affine isoperimetry,” Proc. Lond. Math. Soc. (3), 113, No. 2, 140–162 (2016).
H. Furstenberg and I. Tzkoni, “Spherical functions and integral geometry,” Israel J. Math., 10, 327–338 (1971).
R. J. Gardner, “The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities,” Adv. Math., 216, No. 1, 358–386 (2007).
F. Götze, A. Gusakova, and D. Zaporozhets, “Random affine simplexes,” J. Appl. Probab., 56, No. 1, 39–51 (2019).
E. L. Grinberg, “Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies,” Math. Ann., 291, No. 1, 75–86 (1991).
H. Groemer, “On some mean values associated with a randomly selected simplex in a convex set,” Pacific J. Math., 45, 525–533 (1973).
H. Hadwiger, “Ueber zwei quadratische Distanzintegrale für Eikörper,” Arch. Math. (Basel), 3, 142–144 (1952).
J. F. C. Kingman, “Random secants of a convex body,” J. Appl. Probability, 6, 660–672 (1969).
R. Schneider, “Inequalities for random flats meeting a convex body,” J. Appl. Probab., 22, No. 3, 710–716 (1985).
R. Schneider and W. Weil, Stochastic and Integral Geometry, Springer-Verlag, Berlin (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 505, 2021, pp. 162–171.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Litvak, A.E., Zaporozhets, D.N. Random Section and Random Simplex Inequality. J Math Sci 281, 111–117 (2024). https://doi.org/10.1007/s10958-024-07079-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07079-z