On Grünbaum's problem for symmetric configurations

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 $α_0$. 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 $α_0$ is not attained by any finitely supported distribution.

Disclosure

“e end of Section 3 gives a finitely supported variable with α(X) = log αbin > 0. Hence no finitely supported random variable can attain α0 . □ AI use disclosure Generative AI was used as a writing and editing tool in the preparation of this manuscript. References [1] A. Arman, A. Bondarenko, and A. Prymak, Convex bodies of constant width with exponential illumina”

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2026-08-23T19:05:24.247794Z

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