arXiv subjects
Publications and source records attributed to Barnabás Gárgyán.
We investigate a discrete analogue of the Fisher information. By using a discrete Laplacian operator, we can construct a discrete analogue of the score function and hence of the information. This provides a natural framework for establishing several information-theoretic inequalities in the discrete setting.
We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.
The Laplace--Pólya integral, defined by $J_n(r) = \frac1π\int_{-\infty}^\infty \mathrm{sinc}^n t \cos(rt) \mathrm{d} \, t$, appears in several areas of mathematics. We study this quantity by combinatorial methods; accordingly, our investigation focuses on the values at integer $r$'s. Our main result establishes a lower bound for the ratio $\frac{J_n(r+2)}{J_n(r)}$ which extends and generalises the previous estimates of Lesieur and Nicolas, and provides a natural counterpart to the upper estimate established in our previous work. We derive the statement by purely combinatorial, elementary arguments. As a corollary, we deduce that no subdiagonal central sections of the unit cube are extremal, apart from the minimal, maximal, and the main diagonal sections. We also prove several consequences for Eulerian numbers.
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. 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 $J_n(r) = \int_{-\infty}^\infty \mathrm{sinc}^n t \cdot \cos (rt) \mathrm{d} t$ that is achieved by combinatorial means. This also yields improved bounds for Eulerian numbers of the first kind.