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：受一般概率论的启发，我们称\mathbb{R}^d中的集合X具有“秩k反向性”，如果对于任意k+1个元素q_1,\ldots,q_{k+1}\in X，存在一个从\mathrm{conv} X到k维单纯形\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。当k=1时，这等同于Klee引入的经典（成对）反向性概念。我们考虑Klee关于反向集问题的自然推广：在\mathbb{R}^d中，秩k反向集的最大大小是多少？我们给出了秩k反向集的几何刻画，并借鉴Danzer与Grünbaum最初针对k=1情况发展的论证，证明了一个关于维度的指数级上界。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建立联系，并由此给出了最大大小的下界，该下界同样关于维度指数增长。