Universal optimal configurations for the p-frame potentials
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Given d, N≥ 2 and p∈ (0, ∞] we consider a family of functionals, the p-frame potentials FP_p, N, d, defined on the set of all collections of N unit-norm vectors in R^d. For the special case p=2 and p=∞, both the minima and the minimizers of these potentials have been thoroughly investigated. In this paper, we investigate the minimizers of the functionals FP_p, N, d, by first establishing some general properties of their minima. Thereafter, we focus on the special case d=2, for which, surprisingly, not much is known. One of our main results establishes the unique minimizer for big enough p. Moreover, this minimizer is universal in the sense that it minimizes a large range of energy functions that includes the p-frame potential. We conclude the paper by reporting some numerical experiments for the case d≥ 3, N=d+1, p∈ (0, 2). These experiments lead to some conjectures that we pose.
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