Higher rank antipodality

Naszódi, Márton and Szilágyi, Zsombor and Weiner, Mihály (2023) Higher rank antipodality. arXiv e-prints. ISSN 2331-8422 (Unpublished)

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Abstract

Motivated by general probability theory, we say that the set $X$ in $\textbackslashmathbb{R}⌃d$ is \textbackslashemph{antipodal of rank $k$}, if for any $k+1$ elements $q_1,\textbackslashldots q_{k+1}\textbackslashin X$, there is an affine map from $\textbackslashmathrm{conv} X$ to the $k$-dimensional simplex $\textbackslashDelta_k$ that maps $q_1,\textbackslashldots q_{k+1}$ onto the $k+1$ vertices of $\textbackslashDelta_k$. 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 $\textbackslashmathbb{R}⌃d$? We present a geometric characterization of antipodal sets of rank $k$ and adapting the argument of Danzer and Gr\textbackslash"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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