Motivated by general probability theory, we say that the set $X$ in $\mathbb{R}^d$ is \emph{antipodal of rank $k$}, if for any $k+1$ elements $q_1,\ldots q_{k+1}\in X$, there is an affine map from $\mathrm{conv} X$ to the $k$-dimensional simplex $\Delta_k$ that maps $q_1,\ldots q_{k+1}$ onto the $k+1$ vertices of $\Delta_k$. For $k=1$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank $k$ in $\mathbb{R}^d$? We present a geometric characterization of antipodal sets of rank $k$ and adapting the argument of Danzer and Gr\"unbaum originally developed for the $k=1$ case, we prove an upper bound which is exponential in the dimension. We point out that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension.

翻译：受一般概率论启发，我们称$\mathbb{R}^d$中的集合$X$具有\emph{秩$k$反极点性}，若对任意$k+1$个元素$q_1,\ldots,q_{k+1}\in X$，存在从$\mathrm{conv} X$到$k$维单形$\Delta_k$的仿射映射，将$q_1,\ldots,q_{k+1}$映射到$\Delta_k$的$k+1$个顶点上。当$k=1$时，此定义与Klee引入的（成对）反极点性经典概念一致。我们考虑Klee反极点集问题的如下自然推广：在$\mathbb{R}^d$中，秩$k$反极点集的最大基数是多少？我们给出了秩$k$反极点集合的几何刻画，并通过对Danzer与Grünbaum最初针对$k=1$情形所提出论证方法的适配，证明了该基数存在一个关于维数的指数级上界。同时指出该问题可与计算机科学中关于完美哈希的经典问题相关联，并由此给出最大基数的一个同样呈指数增长的下界。