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Proof of 100 Euro Conjecture

Teng Zhang

Abstract

Dating back to the remarkable 1997 work of S.~M.~Rump \cite{Rum97b}, the celebrated 100 Euro Conjecture has remained open for nearly 30 years. It asserts that for every matrix $A \in \mathbb{R}^{n\times n}$ with $|A|e = ne$, where $|A|$ is the matrix of entrywise absolute values and $e=(1,\dots,1)^T$ is the all-ones vector in $\mathbb{R}^n$, there exists a nonzero vector $x \in \mathbb{R}^n$ such that $|Ax| \ge |x|$ entrywise. In this paper, we confirm this conjecture by means of a finite-dimensional reformulation of Ball's plank theorem. We also record a unified $\ell_p$-escape statement that contains both the cube and Euclidean versions as special cases and yields a weaker form of the 200 Euro Conjecture.

Proof of 100 Euro Conjecture

Abstract

Dating back to the remarkable 1997 work of S.~M.~Rump \cite{Rum97b}, the celebrated 100 Euro Conjecture has remained open for nearly 30 years. It asserts that for every matrix with , where is the matrix of entrywise absolute values and is the all-ones vector in , there exists a nonzero vector such that entrywise. In this paper, we confirm this conjecture by means of a finite-dimensional reformulation of Ball's plank theorem. We also record a unified -escape statement that contains both the cube and Euclidean versions as special cases and yields a weaker form of the 200 Euro Conjecture.
Paper Structure (4 sections, 15 theorems, 72 equations)

This paper contains 4 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Ball's plank theorem in finite-dimensional spaces
  3. The cube-escape theorem on $\ell_\infty^n$
  4. An $\ell_p$-version

Key Result

Theorem 1.4

Let $A=(a_{ij})\in\mathbb{R}^{n\times n}$ and let $r_i=(a_{i1},\ldots,a_{in})$ denote the $i$-th row of $A$. If $\|r_i\|_1\ge nt$ for every $i$ and some $t>0$, then there exists $x\in[-1,1]^n$ such that In particular, $\xi(A)\ge t$; hence $A\in\Omega_n$ when $t=1$.

Theorems & Definitions (30)

  • Conjecture 1.1: Rump
  • Conjecture 1.2: Bünger--Seeger
  • Definition 1.3: BS24
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 20 more