2024/11/06 by Nico Lombardi, Christian Richter, Lombardi, Nico +3
Computer Science · Mathematics · #52A15 #52A20 #52A38 #52A40 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Metric Geometry (math.MG) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2411.03977
openalex publication_date 2024/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Giannopoulos, Hartzoulaki and Paouris asked in \citeGHP whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension 3, using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio.