Non-diagonal critical central sections of the cube. (English) Zbl 1535.51025
Summary: We study the \((n - 1)\)-dimensional volume of central hyperplane sections of the \(n\)-dimensional cube \(Q_n\). 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 \(n \geq 4\), which was shown before by L. Pournin [“Local extrema for hypercube sections”, Preprint, arXiv:2203.15054]. Then, we prove that non-diagonal critical central sections of \(Q_n\) exist in all dimensions at least 4. The crux of both proofs is an estimate on the rate of decay of the Laplace-Pólya integral \(J_n(r) = \frac{1}{\pi} \int_{-\infty}^\infty \operatorname{sinc}^n t \cdot \cos(r t) \mathrm{d} t\) that is achieved by combinatorial means. This also yields improved bounds for Eulerian numbers of the first kind.
MSC:
| 51M25 | Length, area and volume in real or complex geometry |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 05A20 | Combinatorial inequalities |
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