math.MG — Metric Geometry

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

Contributed by Scott Kominers

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.