On certain properties of the Petty space

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.