Antipodal Hadwiger numbers of finite-dimensional Banach spaces

Let $X$ be a finite-dimensional Banach space; we introduce and investigate a natural generalization of the concepts of Hadwiger number $H(X)$ and strict Hadwiger number $H^{\mathrm{\prime }}(X)$. More precisely, we define the antipodal Hadwiger number $H_{\alpha }(X)$ as the largest cardinality of a subset $S\subseteq S_X$, such that $\mathrm{\forall }x\mathrm{\ne }y\in S\mathrm{\exists }f\in B_{X^{\ast }}$ with $$1\le f(x)-f(y)\text{and}f(y)\le f(z)\le f(x)\text{for}z\in S\mathrm{.}$$ The strict antipodal Hadwiger number $H_{\alpha }^{\mathrm{\prime }}(X)$ is defined analogously. We prove that $H_{\alpha }^{\mathrm{\prime }}(X)=4$ for every Minkowski plane and estimate (or in some cases compute) the numbers $H_{\alpha }(X)$ and $H_{\alpha }^{\mathrm{\prime }}(X)$, where $X={\mathrm{\ell }}_p^n,1<p\le +\mathrm{\infty }$ and $n\ge 2$. We also show that the number $H_{\alpha }^{\mathrm{\prime }}(X)$ grows exponentially in $\dim X$.