Higher rank antipodality

TL;DR

This paper generalizes the concept of antipodal sets to higher ranks, characterizes them geometrically, and establishes exponential bounds on their maximum size in Euclidean spaces, linking to problems in computer science and polytope theory.

Contribution

It introduces the notion of rank-$k$ antipodality, provides geometric characterizations, and derives exponential bounds on the size of such sets, connecting to hash functions and neighborly polytopes.

Findings

Upper bounds on size are exponential in dimension.

Lower bounds are exponential, derived from hash function connections.

Connection established between rank-$k$ antipodality and $k$-neighborly polytopes.

Abstract

Motivated by general probability theory, we say that the set $S$ in $\mathbb{R}^d$ is \emph{antipodal of rank $k$}, if for any $k+1$ elements $q_1,\ldots q_{k+1}\in S$, there is an affine map from $\mathrm{conv}(S)$ to the $k$-dimensional simplex $\Delta_k$ that maps $q_1,\ldots q_{k+1}$ bijectively 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 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-$k$ antipodality to $k$-neighborly polytopes, we obtain another upper bound when $k>d/2$.