Higher rank antipodality

• URL: http://arxiv.org/abs/2307.16857v2
• Date: Thu, 30 Nov 2023 13:01:05 GMT
• Title: Higher rank antipodality
• Authors: M\'arton Nasz\'odi and Zsombor Szil\'agyi and Mih\'aly Weiner
• Abstract summary: Motivated by general probability theory, we say that the set $X$ in $mathbbRd$ is emphantipodal of rank $k$. For $k=1$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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