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arXiv · 2611.08198

A diameter-one set in $\mathbb{R}^5$ that cannot be covered by six diameter-one balls

A. Arman·A. Bondarenko·A. Prymak·D. Radchenko
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Abstract

We construct a compact set of diameter $1$ in $\mathbb{R}^5$ which cannot be covered by six Euclidean balls of diameter $1$. Its convex hull is a full-dimensional convex body satisfying the same property. Moreover, a finite subset of diameter $1$ cannot be covered by six closed balls even when their radius is increased to $0.50001$. The compact set is the quadratic image of the unit sphere in the space of symmetric $3\times3$ matrices of trace zero. A hypothetical cover leads to twelve spherical ellipses covering ${\mathbb S}^2$ and arranged in six antipodal pairs. Using Euler's formula, we replace them by twelve pentagonal regions contained in the ellipses. An area estimate then gives a contradiction.

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A. Arman, A. Bondarenko, A. Prymak, D. Radchenko. 2026-11-05. A diameter-one set in $\mathbb{R}^5$ that cannot be covered by six diameter-one balls. https://arxiv.org/abs/2611.08198

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