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

We prove Rump’s 100-euro conjecture by deriving a weighted affine escape theorem from Ball’s plank theorem in [Invent. Math. $\mathbf{\text{104}}$ (1991)]. More precisely, let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ and let $A\in {\mathbb{K}}^{n\times n}$. For every $1\le p\le \mathrm{\infty }$, we obtain an ${\mathrm{\ell }}_p$-escape principle controlled by the row ${\mathrm{\ell }}_q$-norms of $A$. Its cube case shows that $\mathrm{\mid }A\mathrm{\mid }e=ne$, where $e$ is the all-one vector, implies the existence of a nonzero vector $x$ satisfying $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ and $\mathrm{\mid }Ax\mathrm{\mid }\ge e\ge \mathrm{\mid }x\mathrm{\mid }$, thereby settling the conjecture. As a consequence, we prove the global comparison ${\rho }_0(\mathrm{\mid }A\mathrm{\mid })\le n{\rho }_{\mathbb{K}}(A)$,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 ${\mathrm{\ell }}_{\mathrm{\infty }}$-escape principle sharpens Rump’s result in [SIAM Rev. $\mathbf{\text{41}}$ (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 Bünger.