Abstract
We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1]n. We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of ℝn onto a k-dimensional subspace that maximizes the volume of the intersection. We find the optimal upper bound on the volume of a planar section of the cube [−1, 1]n, n ≥ 2.
References
[1] Arseniy Akopyan, Alfredo Hubard, and Roman Karasev, Lower and upper bounds for the waists of different spaces, Topological Methods in Nonlinear Analysis 53 (2019), no. 2, 457–490.Search in Google Scholar
[2] Keith Ball, Cube slicing in ℝn, Proceedings of the American Mathematical Society (1986), 465–473.10.1090/S0002-9939-1986-0840631-0Search in Google Scholar
[3] Keith Ball, Volumes of sections of cubes and related problems, Geometric aspects of functional analysis (1989), 251–260.10.1007/BFb0090058Search in Google Scholar
[4] Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor, A probabilistic approach to the geometry of the lnp-ball, The Annals of Probability 33 (2005), no. 2, 480–513.Search in Google Scholar
[5] Stefano Campi and Paolo Gronchi, On volume product inequalities for convex sets, Proceedings of the American Mathematical Society 134 (2006), no. 8, 2393–2402.Search in Google Scholar
[6] Alexandros Eskenazis, Piotr Nayar, and Tomasz Tkocz, Gaussian mixtures: entropy and geometric inequalities, The Annals of Probability 46 (2018), no. 5, 2908–2945.Search in Google Scholar
[7] Alexandros Eskenazis, On Extremal Sections of Subspaces of Lp, Discrete & Computational Geometry (2019), 1–21.Search in Google Scholar
[8] Peter M. Gruber, Convex and discrete geometry, Springer Berlin Heidelberg, 2007.Search in Google Scholar
[9] Grigory Ivanov, On the volume of projections of cross-polytope, arXiv preprint arXiv:1808.09165 (2018).Search in Google Scholar
[10] Grigory Ivanov, Tight frames and related geometric problems, accepted for publication in Canadian Mathematical Bulletin.Search in Google Scholar
[11] Grigory Ivanov, On the volume of the John–Löwner ellipsoid, Discrete & Computational Geometry 63, 455–459 (2020).10.1007/s00454-019-00071-4Search in Google Scholar
[12] Hermann König and Alexander Koldobsky, On the maximal measure of sections of the n-cube, Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, Contemp. Math 599 (2012), 123–155.10.1090/conm/599/11907Search in Google Scholar
[13] Alexander Koldobsky, An application of the Fourier transform to sections of star bodies, Israel Journal of Mathematics 106 (1998), no. 1, 157–164.Search in Google Scholar
[14], Fourier analysis in convex geometry, no. 116, American Mathematical Soc., 2005.Search in Google Scholar
[15] Mathieu Meyer and Alain Pajor, Sections of the unit ball of lnp, Journal of Functional Analysis 80 (1988), no. 1, 109–123.Search in Google Scholar
[16] Fedor L. Nazarov and Anatoliy N. Podkorytov, Ball, Haagerup, and distribution functions, Complex analysis, operators, and related topics, Springer, 2000, pp. 247–267.10.1007/978-3-0348-8378-8_21Search in Google Scholar
[17] Jeffrey Vaaler, A geometric inequality with applications to linear forms, Pacific Journal of Mathematics 83 (1979), no. 2, 543–553.Search in Google Scholar
[18] Chuanming Zong, The cube: A window to convex and discrete geometry, Cambridge Tracts in Mathematics, vol. 168, Cambridge University Press, 2006.Search in Google Scholar
[19] Artem Zvavitch, Gaussian measure of sections of dilates and translations of convex bodies, Advances in Applied Mathematics 41 (2008), no. 2, 247–254.Search in Google Scholar
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