On prescribing total preorders and linear orders to pairwise distances of points in Euclidean space
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We show that any total preorder on a set with n2 elements coincides with the order on pairwise distances of some point collection of size n in ℝ^n-1. For linear orders, a collection of n points in ℝ^n-2 suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for linear orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.
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