Higher rank antipodality

TL;DR

The paper generalizes antipodality to rank $k$ in $\mathbb{R}^d$, defining $A(d,k)$ as the maximum size of a rank-$k$ antipodal set and establishing both upper and lower bounds. An exponential-in-$d$ volume-based upper bound is obtained via a Danzer–Grünbaum–style dilation argument, and a geometric link to $k$-neighborly polytopes yields a sharp bound $A(d,k)=d+1$ when $k>d/2$. On the constructive side, the authors connect rank-$k$ antipodality to perfect $(b,k+1)$-hash codes, showing that hashing-based constructions produce exponential lower bounds in dimension, and thus there is a persistent gap between upper and lower bounds for $k>2$. They further relate the problem to generalized probability theory (GPT), where joint distinguishability corresponds to affine maps to simplices, and discuss potential directions, including relaxing joint distinguishability and exploiting neighborliness to bridge geometry and combinatorics. Overall, the work bridges convex geometry, combinatorics, and information-theoretic ideas to understand the limits and possibilities of higher-rank antipodality and its computational implications.

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 $Δ_k$ that maps $q_1,\ldots q_{k+1}$ bijectively onto the $k+1$ vertices of $Δ_k$. 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 Grünbaum 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 $k>d/2$.

Theorems & Definitions (14)

• Theorem 1: A general upper bound on the size of a rank $k$ antipodal set
• Theorem 2: Upper bound on $A(d,k)$ using $k$-neighborly polytopes
• Theorem 3: Hashes yield rank $k$ antipodal sets
• Proposition 4: Characterization of joint antipodality in terms of dilates
• Corollary 5
• proof : Proof of Theorem \ref{['thm:upperbound']}
• Lemma 6: Separation of multiple convex sets
• proof : Proof of Lemma \ref{['lem:barany']}
• Lemma 7: Non-overlapping dilates of a simplex
• proof