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Breaking the Fourth Wall: The Planar Centroid Banach--Mazur Diameter Is Strictly Less than $4$
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≈ 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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BibTeX
@misc{kominers2026breaking,
title = {{Breaking the Fourth Wall: The Planar Centroid Banach--Mazur Diameter Is Strictly Less than $4$}},
author = {Scott Duke Kominers},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13952},
eprint = {2610.13952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13952},
}
RIS
TY - UNPB
TI - Breaking the Fourth Wall: The Planar Centroid Banach--Mazur Diameter Is Strictly Less than $4$
AU - Scott Duke Kominers
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13952
UR - https://arxiv.org/abs/2610.13952
AB - 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≈ 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.
ER -