Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.09542

A note on Erdős matrices and Marcus\unicode{x2013}Ree inequality

Aman Kushwaha·Raghavendra Tripathi
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1959, Marcus and Ree proved that any bistochastic matrix $A$ satisfies $Δ_n(A):= \max_{σ\in S_n}\sum_{i=1}^{n}A(i, σ(i))-\sum_{i, j=1}^n A(i, j)^2 \geq 0$. Erdős asked to characterize the bistochastic matrices satisfying $Δ_n(A)=0$. This problem remains largely open, and very recently, a complete list of such matrices was obtained in dimension $n=3$ by Bouthat, Mashreghi, and Morneau-Guérin. Soon after, Tripathi proved that there were only finitely many such matrices in any dimension $n$. In this paper, we continue the investigation initiated in these two works. We characterize all $4\times 4$ bistochastic matrices satisfying $Δ_4(A)=0$. Furthermore, we show that for $n\geq 3$, $Δ_n(A)=α$ has uncountably many solutions when $α\in (0, (n-1)/4)$. This answers a question raised in [Tripathi, R., $\textit{Some observations on Erdős matrices}$, Linear Algebra and Its Applications 708 (2025)]. We also extend the Marcus\unicode{x2013}Ree inequality to infinite bistochastic arrays and bistochastic kernels. Our investigation into $4\times 4$ Erdős matrices also leads to several intriguing questions of independent interest. We propose several questions and conjectures and present numerical evidence for them.

Explore related subjects

math.MGmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Aman Kushwaha, Raghavendra Tripathi. 2025-07-20. A note on Erdős matrices and Marcus\unicode{x2013}Ree inequality. https://doi.org/10.1016/j.laa.2025.07.012

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$β$-Uniform Convexity and Divisible Domains

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-Ichi Ohta as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.

math.MG↗
arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗
arXiv2026

Minimal central slices of the regular simplex

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.

math.MG↗