Grigory Ivanov, Igor Tsiutsiurupa
January 29, 2021
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 ℝ n 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.