On certain properties of the Petty space

We study some touching properties of the three-dimensional Petty space $X=({\mathrm{\ell }}_2^2\oplus \mathbb{R}{)}_1$. In particular we give an estimation of its Hadwiger number and also show that its equilateral subsets $A$ of maximum cardinality (i.e. $\mathrm{\mid }A\mathrm{\mid }=e(X)$) do not have a center.