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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2603.07423 |
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| _version_ | 1866910134170550272 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | 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 $ρ_0(|A|)\le n\,ρ_{\mathbb{K}}(A)$,where $ρ_{\mathbb{K}}$ 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 Bünger. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07423 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Plank theorems, Gaussian probabilistic estimates and Rump's 100 Euro conjecture Zhang, Teng Functional Analysis 46B20, 15A18, 15A42, 52A40 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 $ρ_0(|A|)\le n\,ρ_{\mathbb{K}}(A)$,where $ρ_{\mathbb{K}}$ 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 Bünger. |
| title | Plank theorems, Gaussian probabilistic estimates and Rump's 100 Euro conjecture |
| topic | Functional Analysis 46B20, 15A18, 15A42, 52A40 |
| url | https://arxiv.org/abs/2603.07423 |