Plank theorems, Gaussian probabilistic estimates and Rump's 100 Euro conjecture

TL;DR

This paper proves Rump's 100-euro conjecture using a weighted affine escape theorem derived from Ball's plank theorem, establishing sharp spectral radius bounds and improving previous estimates with probabilistic methods.

Contribution

It introduces a new $ ext{ell}_p$-escape principle, proves the conjecture, and sharpens spectral radius estimates, extending Rump's results with probabilistic and geometric techniques.

Findings

Proved Rump's 100-euro conjecture.

Established sharp spectral radius bounds for matrices.

Used Gaussian probabilistic estimates to analyze complex analogues.

Abstract

We prove Rump's 100-euro conjecture by deriving a weighted affine escape theorem from Ball's plank theorem in [Invent. Math. \textbf{104} (1991)]. More precisely, let $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ and let $A\in\mathbb{K}^{n\times n}$. For every $1\le p\le \infty$, we obtain an $\ell_p$-escape principle controlled by the row $\ell_q$-norms of $A$. Its cube case shows that $|A|e=ne$, where $e$ is the all-one vector, implies the existence of a nonzero vector $x$ satisfying $\|x\|_{\infty}\le 1$ and $|Ax|\ge e\ge |x|$, thereby settling the conjecture. As a consequence, we prove the global comparison $\rho_0(|A|)\le n\,\rho_{\mathbb{K}}(A)$,where $\rho_{\mathbb{K}}$ denotes the sign-real or complex spectral radius, respectively. This is the sharp form of Rump's Perron--Frobenius-type estimate, with the factor $3+2\sqrt{2}$ removed. Moreover, our $\ell_\infty$-escape principle sharpens Rump's result in [SIAM Rev. \textbf{41} (1999)] on the relation between the entrywise distance to singularity of a matrix and its entrywise Bauer--Skeel condition number. Finally, we also investigate the weaker Euclidean row condition, including sharp quantitative bounds and counterexamples to possible strengthenings. In particular, we use Gaussian probabilistic estimates to establish a complex analogue of a conjecture of B\"unger.