On certain properties of the Petty space

TL;DR

This paper investigates specific geometric properties of the three-dimensional Petty space, including bounds on its Hadwiger number and characteristics of its largest equilateral subsets, revealing they lack a center.

Contribution

It provides new estimations of the Hadwiger number and demonstrates that maximal equilateral subsets in the Petty space do not possess a center.

Findings

Estimated the Hadwiger number of the Petty space.

Showed that maximal equilateral subsets lack a center.

Analyzed touching properties of the Petty space.

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.