On the Hadwiger numbers of starlike disks

TL;DR

This paper improves the upper bound on the Hadwiger number for starlike disks from 75 to 35, advancing understanding of how many nonoverlapping translates can touch such disks.

Contribution

It establishes a tighter upper bound of 35 for the Hadwiger number of starlike disks, refining previous bounds and exploring related topological conditions.

Findings

Hadwiger number of starlike disks is at most 35

Hadwiger number for certain topological disks is 6 or 8

Improves previous upper bound of 75

Abstract

The Hadwiger number $H(J)$ of a topological disk $J$ in $\Re^2$ is the maximal number of pairwise nonoverlapping translates of $J$ that touch $J$. It is well known that for a convex disk, this number is six or eight. A conjecture of A. Bezdek., K. and W. Kuperberg says that the Hadwiger number of a starlike disk is at most eight. A. Bezdek proved that this number is at most seventy five for any starlike disk. In this note, we prove that the Hadwiger number of a starlike disk is at most thirty five. Furthermore, we show that the Hadwiger number of a topological disk $J$ such that $(\conv J) \setminus J$ is connected, is six or eight.