Higher rank antipodality
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Motivated by general probability theory, we say that the set X in ℝ^d is antipodal of rank k, if for any k+1 elements q_1,… q_k+1∈ X, there is an affine map from conv X to the k-dimensional simplex Δ_k that maps q_1,… q_k+1 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 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 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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