HEXAGONbeta

Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less than 44

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometry

Contributed by Scott Kominers ↗

SubmitterScott Kominers

Version 1 / Sep 28, 2026 / CC BY-SA 4.0

Abstract

We prove that the centroid Banach–Mazur diameter of planar convex bodies is strictly less than 44, (slightly) improving the prior upper bound 69/17≈4.0588269/17\approx 4.05882 due to Lassak. The proof combines a sharp covariance-ellipse sandwich, whose equality cases in any single direction occur only for triangles, with a compactness argument: an extremal pair at distance 44 would consist of two triangles, which are linearly equivalent and hence must in fact be at distance 11. The resulting gap below 44 is uniform but nonquantitative.

Provenance statement

I used LLMs to assist with computations, analysis, and synthesis in the preparation of this article, especially GPT-5.6 Sol and Claude Fable 5 (both accessed in part via Poe with the support of Quora, where I am an advisor). In particular, GPT-5.6 Sol identified an analytic shortcut that substantially simplified a prior version of the argument.

Tools used

OpenAI
GPTVersion 5.6 Sol
Anthropic
Claude FableVersion 5
Quora
Poe

References

  1. A. S. Besicovitch, Measure of asymmetry of convex curves , Journal of the London Mathematical Society s1-23 (1948), no. 3, 237--240.
  2. Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of isotropic convex bodies , Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014.
  3. Matthieu Fradelizi and Olivier Gu \'e don, The extreme points of subsets of \(s\)-concave probabilities and a geometric localization theorem , Discrete & Computational Geometry 31 (2004), no. 2, 327--335.
  4. Matthieu Fradelizi, Grigoris Paouris, and Carsten Sch \"u tt, Simplices in the Euclidean ball , Canadian Mathematical Bulletin 55 (2012), no. 3, 498--508.
  5. Branko Gr \"u nbaum, Measures of symmetry for convex sets , Convexity (Victor Klee, ed.), Proceedings of Symposia in Pure Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1963, pp. 233--270.
  6. Ravi Kannan, L \'a szl \'o Lov \'a sz, and Mikl \'o s Simonovits, Isoperimetric problems for convex bodies and a localization lemma , Discrete & Computational Geometry 13 (1995), no. 3--4, 541--559.
  7. Marek Lassak, The centroid Banach--Mazur distance between the parallelogram and the triangle , Journal of Convex Analysis 31 (2024), no. 1, 51--58.
  8. , Estimation of the centroid Banach--Mazur distance between planar convex bodies , Ukrainian Mathematical Journal 76 (2024), no. 5, 872--879.
  9. , Position of the centroid of a planar convex body , Aequationes Mathematicae 98 (2024), no. 3, 687--695.
  10. V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed \(n\)-dimensional space , Geometric Aspects of Functional Analysis: Israel Seminar ( GAFA ) 1987--88 (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Mathematics, vol. 1376, Springer, Berlin, Heidelberg, 1989, pp. 64--104.

Version history

  1. v1Initial depositCurrentSep 28, 2026