On a Dowker-Type Problem for Convex Disks with Almost Constant Curvature.

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body , the areas of the maximum (resp. minimum) area convex -gons inscribed (resp. circumscribed) in is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex -gons by disk--gons, obtained as the intersection of closed Euclidean unit disks. It has been proved recently that if is the unit disk of a normed plane, then the same properties hold for the area of --gons circumscribed about a -convex disk and for the perimeters of --gons inscribed or circumscribed about a -convex disk , but for a typical origin-symmetric convex disk with respect to Hausdorff distance, there is a -convex disk such that the sequence of the areas of the maximum area --gons inscribed in 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. [ABSTRACT FROM AUTHOR]