2019/05/31 by Florian Besau, Steven Hoehner, Gil Kur · 8 citations
Mathematics · #Ball (mathematics) #Convex body #Convex polytope #Euclidean distance #Euclidean geometry #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #Polytope #Regular polygon #Unit sphere #math.MG #msc:52A20 #msc:52A22 #msc:52A27 #msc:52A39 #msc:52B05 #msc:52B11
paper · pdf · doi:10.1093/imrn/rnz277
published in International Mathematics Research Notices 2021(22), 17456-17513 (Oxford University Press) · 44 pages, 1 figure. To appear in International Mathematics Research Notices
openalex created_date 2019/05/29 · openalex publication_date 2019/10/01 · arxiv created 2020/02/27 · arxiv updated 2020/03/02 · openalex updated_date 2026/08/06
Abstract We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.