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arXiv · 2607.22032

On Gr\"unbaum's problem for symmetric configurations

Andrii Arman·Andriy Bondarenko·Andriy Prymak·Danylo Radchenko
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Abstract

Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $\alpha_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $\alpha_0$ is not attained by any finitely supported distribution.

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Andrii Arman, Andriy Bondarenko, Andriy Prymak, Danylo Radchenko. 2026-07-24. On Gr\"unbaum's problem for symmetric configurations. https://arxiv.org/abs/2607.22032

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