Dowker-type theorems for disk-polygons in normed planes

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.

Figures (5)

• Figure 1: An illustration for the proof of Lemma \ref{['lem:quadrangle']}, with a Euclidean disk as $C$.
• Figure 2: An illustration for the notation in the proof of Theorems \ref{['thm:Dowker1']} and \ref{['thm:Dowker2']}. The left panel shows the $C$-spindle $[\Gamma(\varphi_1), \Gamma(\varphi_2)]_C$ (the region with little crosses), and the boundaries of the support disks $C(\varphi_1)$, $C(\varphi_2)$ (dashed lines). The right panel shows the regions $r(\varphi_1,\varphi_2)$ (with horizontal stripes) and $R(\varphi_1,\varphi_2)$ (with dots).
• Figure 3: Notation for the proof of Theorems \ref{['thm:Dowker1']} and \ref{['thm:Dowker2']} for $\hat{P}_n^C(K)$.
• Figure 4: The hexagon $H$ and the octagon $K_1$. In the illustration, $t=10$.
• Figure 5: The $C_4$-convex disk $K_4$ with $t=10$ and $s=20$.

Theorems & Definitions (38)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Remark 2
• Remark 3