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PreprintCC BY 4.0HAL

On Grünbaum's problem for symmetric configurations

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

Let gn be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in ℝn. 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 α0. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound gn≥ (1.160235457…−o(1))n, improving the previous best bound (2/√3−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 α0 is not attained by any finitely supported distribution.

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Cite

BibTeXRIS
BibTeX
@misc{arman2026grünbaums,
  title = {{On Grünbaum's problem for symmetric configurations}},
  author = {Andrii Arman and Andriy Bondarenko and Andriy Prymak and Danylo Radchenko},
  howpublished = {HAL},
  year = {2026},
  month = jul,
  url = {https://hal.science/hal-05851861v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Grünbaum's problem for symmetric configurations
AU  - Andrii Arman
AU  - Andriy Bondarenko
AU  - Andriy Prymak
AU  - Danylo Radchenko
PY  - 2026
DA  - 2026/07/24
PB  - HAL
UR  - https://hal.science/hal-05851861v1
AB  - 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 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≥ (1.160235457…−o(1))^n, improving the previous best bound (2/√3−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.
ER  -