Antipodal Hadwiger numbers of finite-dimensional Banach spaces

Abstract

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

19 pages, 1 figure. To appear in Beiträge zur Algebra und Geometrie

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• The geometry of Minkowski spaces -- a survey. Part I
• A lower bound for the equilateral number of normed spaces
• Combinatorial distance geometry in normed spaces