Abstract Motivated by general probability theory, we say that the set in is antipodal of rank , if for any elements , there is an affine map from to the ‐dimensional simplex that maps bijectively onto the vertices of . For , 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 in ? We present a geometric characterization of antipodal sets of rank and adapting the argument of Danzer and Grünbaum originally developed for the case, we prove an upper bound which is exponential in the dimension. We show 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. By connecting rank‐ antipodality to ‐neighborly polytopes, we obtain another upper bound when .
Naszódi et al. (Mon,) studied this question.