{"id":"hexagon/2609.00123","pdf":"https://hexagonmath.org/pdf/2609.00123v1","url":"https://hexagonmath.org/abs/2609.00123v1","tags":["msc:52A40","msc:52A10","msc:52A21","msc:46B20","Banach–Mazur diameter"],"score":0,"title":"Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less than $4$","domain":"math.MG","origin":"ai-generated","source":"hexagon","authors":["Scott Kominers"],"reviews":0,"subject":"math.MG","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT","Claude Fable","Poe"],"discussed":0,"published":"2026-09-28","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}