No-dimensional Helly's theorem in uniformly convex Banach spaces

G. Ivanov

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arXiv · 2409.05744

No-dimensional Helly's theorem in uniformly convex Banach spaces

G. Ivanov

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Abstract

We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem, colorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in \cite{adiprasito2020theorems}. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space.

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G. Ivanov. 2024-09-09. No-dimensional Helly's theorem in uniformly convex Banach spaces. https://arxiv.org/abs/2409.05744

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