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

Bushra Basit, Zsolt Lángi

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 $C$.

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

Abstract

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 .
Paper Structure (6 sections, 6 theorems, 34 equations, 3 figures)

This paper contains 6 sections, 6 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:old']}
  4. Deriving a formula for the mixed partial derivative $(\partial_s \partial_t A)(\bar{s},\bar{t})$
  5. Determining the sign of the derivative if $C$ is a Euclidean disk
  6. Proof of Theorem \ref{['thm:counter']}

Key Result

Theorem 1

For any $n \geq 4$ and spindle convex disk $K$, we have $\hat{a}_{n-1}^{B^2}(K) + \hat{a}_{n+1}^{B^2}(K) \leq 2 \hat{a}_n^{B^2}(K)$.

Figures (3)

  • Figure 1: An illustration for the Quadrangle Property. According to it, the area of the dashed region is not less than the area of the dotted region.
  • Figure 2: The vectors described in Remark \ref{['rem:order']}. The $C$-spindle $[p,q]_C$ is illustrated by a dashed region.
  • Figure 3: An illustration for the notation in the proof of Theorem \ref{['thm:old']}. The region $R$ is denoted by skew lines.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Lemma 1
  • Definition 5
  • Proposition 1
  • ...and 7 more