Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access January 29, 2021

On the Volume of Sections of the Cube

Grigory Ivanov and Igor Tsiutsiurupa EMAIL logo


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.

MSC 2010: 52A38; 49Q20; 52A40; 15A45


[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 inn, 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

Received: 2020-04-17
Accepted: 2020-12-14
Published Online: 2021-01-29

© 2021 Grigory Ivanov et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 30.1.2023 from
Scroll Up Arrow