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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

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.

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

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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.

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