Naszódi, Márton and Szilágyi, Zsombor and Weiner, Mihály (2023) Higher rank antipodality. arXiv eprints. ISSN 23318422 (Unpublished)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 wellstudied 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.
Item Type:  Article 

Additional Information:  arXiv:2307.16857 quantph 
Uncontrolled Keywords:  52C17, Computer Science  Information Theory, Mathematics  Metric Geometry, Quantum Physics 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika > QA73 Geometry / geometria 
Depositing User:  Dr. Márton Naszódi 
Date Deposited:  27 Sep 2023 11:35 
Last Modified:  27 Sep 2023 11:35 
URI:  http://real.mtak.hu/id/eprint/175280 
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