Non-diagonal critical central sections of the cube

Ambrus, Gergely and Gárgyán, Barnabás (2024) Non-diagonal critical central sections of the cube. ADVANCES IN MATHEMATICS, 441. ISSN 0001-8708

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Abstract

We study the (n − 1)-dimensional volume of central hyperplane sections of the n-dimensional cube Qn . Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular to the main diagonal of the cube is strictly locally maximal for every n ≥ 4, which was shown before by L. Pournin [27]. Then, we prove that non-diagonal critical central sections of Qn exist in all dimensions at least 4. The crux of both proofs is an estimate on the rate of decay of the Laplace-Pólya inte- gral Jn (r ) = R ∞ −∞ sincn t · cos(r t ) dt that is achieved by combinatorial means. This also yields improved bounds for Eulerian numbers of the first kind.

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