On a Dowker-type problem for convex disks with almost constant curvature

TL;DR

This paper explores a classical geometric problem related to convex polygons inscribed or circumscribed about convex disks, focusing on how different topologies affect the concavity or convexity properties of area and perimeter sequences.

Contribution

It investigates the impact of topology induced by surface area measure on Dowker-type properties for convex disks with almost constant curvature.

Findings

Concavity of area sequences can fail under certain topologies.

Surface area measure topology provides new insights into convex disk properties.

Results extend classical Dowker theorems to broader contexts.

Abstract

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 $C$.