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arXiv · 2610.13952

Breaking the Fourth Wall: The Planar Centroid Banach--Mazur Diameter Is Strictly Less than $4$

Scott Duke Kominers
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Abstract

We prove that the centroid Banach--Mazur diameter of planar convex bodies is strictly less than $4$, (slightly) improving the prior upper bound $69/17\approx 4.05882$ 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 $4$ would consist of two triangles, which are linearly equivalent and hence must in fact be at distance $1$. The resulting gap below $4$ is uniform but nonquantitative.

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Scott Duke Kominers. 2026-08-11. Breaking the Fourth Wall: The Planar Centroid Banach--Mazur Diameter Is Strictly Less than $4$. https://arxiv.org/abs/2610.13952

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