On certain properties of the Petty space

TL;DR

The paper investigates touching properties of the three-dimensional Petty space $X=(\ell_2^2 \oplus \mathbb{R})_1$. It provides bounds on the Hadwiger number $H(X)$ by constructing a $1^+$-separated subset of $S_X$ with 14 points to obtain $H'(X)\ge 14$, and by a stratified angular-distance analysis to obtain $H(X)\le 16$, with a conjecture that $H(X)=H'(X)=14$. It also determines that the maximal equilateral set size is $e(X)=5$ and proves that no 5-point $1$-equilateral configuration has a center; the paper also discusses 4-point maximal equilateral sets, including explicit examples with and without centers. The results sharpen understanding of kissing/ touching configurations in Petty space and suggest the true Hadwiger number is 14.

Abstract

We study some touching properties of the three-dimensional Petty space $X=(\ell_2^2 \oplus \mathbb{R})_1$. In particular we give an estimation of its Hadwiger number and also show that its equilateral subsets $A$ of maximum cardinality (i.e. $|A|=e(X)$) do not have a center.

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Theorems & Definitions (41)

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