Dowker-type theorems for disk-polygons in normed planes

TL;DR

This paper extends Dowker's theorems to disk-polygons in normed planes, showing that certain area and perimeter properties hold for these shapes, but also identifying cases where the theorems fail.

Contribution

It generalizes Dowker's theorems to $C$-$n$-gons in normed planes and investigates their validity and limitations.

Findings

Dowker's theorem holds for areas and perimeters of circumscribed $C$-$n$-gons.

Dowker's theorem holds for the perimeters of inscribed $C$-$n$-gons.

The theorem fails for the areas of inscribed $C$-$n$-gons in typical symmetric convex bodies.

Abstract

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body $K$ in the Euclidean plane, 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, the Euclidean plane by an arbitrary normed plane, or convex $n$-gons by disk-$n$-gons, obtained as the intersection of $n$ closed Euclidean unit disks. The aim of our paper is to investigate these problems for $C$-$n$-gons, defined as intersections of $n$ translates of the unit disk $C$ of a normed plane. In particular, we show that Dowker's theorem remains true for the areas and the perimeters of circumscribed $C$-$n$-gons, and the perimeters of inscribed $C$-$n$-gons. We also show that in the family of origin-symmetric plane convex bodies, for a typical element $C$ with respect to Hausdorff distance, Dowker's theorem for the areas of inscribed $C$-$n$-gons fails.