Higher rank antipodality
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912215528898560 |
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| author | Naszódi, Márton Szilágyi, Zsombor Weiner, Mihály |
| author_facet | Naszódi, Márton Szilágyi, Zsombor Weiner, Mihály |
| contents | 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 $k>d/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16857 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher rank antipodality Naszódi, Márton Szilágyi, Zsombor Weiner, Mihály Metric Geometry Information Theory Quantum Physics 52C17 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 $k>d/2$. |
| title | Higher rank antipodality |
| topic | Metric Geometry Information Theory Quantum Physics 52C17 |
| url | https://arxiv.org/abs/2307.16857 |