HEXAGONbeta

Optimizing a cube–ball lower bound for the reverse Brunn–Minkowski constant

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

Contributed by Ricardo Pinho ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

We optimize a single term in the cube–Euclidean-ball construction of V. Patankar to obtain C_70 >= exp(1/(2e)) > 1.20194336847. The strongest public proposal located and reproduced in our dated source audit is Patankar's 1.201927, whose specified witness evaluates near 1.2019271425725. A separate finite-dimensional witness certifies 1.20194336. Two separately written arithmetic implementations, using Arb and rational logarithm/Stirling bounds respectively, verify the numerical comparisons and reject corrupted or excessive claims. The geometric reduction and passage to the limit are proved here. This unrefereed preprint is not peer reviewed and not formally verified; exhaustive priority is not asserted.

Provenance statement

Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared the certificate, checkers and publication materials. This does not claim a line-by-line human verification of the proof or software. Two separately written arithmetic checkers (Arb ball arithmetic and rational logarithm/Stirling bounds; written by the same AI agent, without importing one another) verify the numerical comparisons, and corrupted or excessive claims are rejected as intended. Repository: https://github.com/rpinho/c70-certificate (commit 680e074bb50929734bfd76484d716cd3e85827f0); the certificate and both checkers are also included in anc/.

Tools used

OpenAI
CodexVersion GPT-6.1 Sol
FLINT
Arb ball arithmetic (python-flint)Version 0.9.0

References

  1. T. Tao and contributors, Optimization Constants in Mathematics , entry 70a, https://teorth.github.io/optimizationproblems/constants/70a.html.link
  2. V. Patankar, Rigorous lower bounds for constants 16a, 20a, 20c and 70a, and an upper bound for 23b , public code and proofs, 2026, source tree c7c811aa06430527cccd5f992e41cd38dd9548ca , https://github.com/vidigoat/optimization-records, Tao-table PR226.link
  3. C. Toledo, Five pull requests to the constants registry, decided where decidable , replay dated 16 October 2026, https://carlostoledo.co/reports/optconst-queue.html.link
  4. H. Robbins, A remark on Stirling's formula, Amer. Math. Monthly 62 (1955), 26--29.

Version history

  1. v1Submitted by Ricardo PinhoInitial depositCurrentOct 23, 2026