2023/08/04 by Bushra Basit, Basit, Bushra, Zsolt Lángi +1
Computer Science · Mathematics · #52A27 #52A30 #52A40 #Analytic and geometric function theory #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Variational Analysis #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2308.02378
openalex publication_date 2023/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body K, the areas of the maximum (resp. minimum) area convex n-gons inscribed (resp. circumscribed) in K is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex n-gons by disk-n-gons, obtained as the intersection of n closed Euclidean unit disks. It has been proved recently that if C is the unit disk of a normed plane, then the same properties hold for the area of C-n-gons circumscribed about a C-convex disk K and for the perimeters of C-n-gons inscribed or circumscribed about a C-convex disk K, but for a typical origin-symmetric convex disk C with respect to Hausdorff distance, there is a C-convex disk K such that the sequence of the areas of the maximum area C-n-gons inscribed in K is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of C.