HEXAGONbeta

An exponential improvement for Borsuk's problem

Primarily AI-generated textHuman understanding: all partsmath.MG — Metric Geometry

Contributed by Andriy Prymak ↗

Version 1 / Oct 08, 2026 / CC BY 4.0

Abstract

We present a method for improving the classical exponential base 3/2\sqrt{3/2} in the upper bound for Borsuk's partition number. The emphasis is on explaining the method rather than optimizing the bound. The problem is reduced to constructing directions which make an acute angle with every member of a prescribed spherical set. These play the same role as the illuminating directions in Schramm's method. The geometric ingredients are an antipodal pairing, a lift by one coordinate, and reflection in a self-dual cone. This reflection folds Euclidean space into the cone. A translation opposite to the horizontal part of its Gaussian mean makes the required supporting inequalities hold with high probability. The Gaussian Poincar\'e inequality controls the mean, and the formula for translating a Gaussian density bounds the measure cost of this construction.

Provenance statement

The main proof arguments were primarily generated using ChatGPT-6 Astra with discussion and prompts by Andrii Arman, Andriy Bondarenko, Andriy Prymak and Danylo Radchenko. Andriy Prymak curates this note but does not claim authorship of its mathematical results. The primary context provided to AI was the study of small spherical bodies of constant width by Hall, Prymak and Sujsuntinukul arXiv:2606.30960v2. Sources of the key proof ingredients are credited in the text.

Tools used

OpenAI
ChatGPTVersion Astra

References

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  3. A. Hall, A. Prymak, and C. Sujsuntinukul, Exponential bounds for the spherical Blaschke--Lebesgue problem , 2026. https://arxiv.org/abs/2606.30960 arXiv:2606.30960 .arXiv
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Version history

  1. v1Submitted by Andriy PrymakInitial depositCurrentOct 08, 2026