For any positive real number @math , the @math -frame potential of @math unit vectors @math is defined as @math . In this paper, we focus on the special case @math and establish the unique minimizer of @math for @math . Our results completely solve the minimization problem of @math -frame potential when @math , which confirms a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou.
However, when @math , not much is known except few special cases. In @cite_12 , Ehler and Okoudjou solved the simplest case where @math and @math and also proved that the minimizer of the @math -frame potential is exactly @math copies of an orthonormal basis if @math where @math is a positive integer. In @cite_4 , Glazyrin provided a lower bound for any @math : but the condition under which the equality holds is very harsh. In @cite_13 , Chen, Gonzales, Goodman, Kang and Okoudjou considered this special case where @math . Particularly, numerical experiments in @cite_13 show that the set @math , which is called lifted ETF, seems to be the minimizer of the @math -frame potential where @math is an integer depending on @math . Here, @math is defined as a set of @math unit vectors in @math satisfying . . Note that @math actually forms an ETF in some subspace @math with dimension @math and the rest of @math vectors form an orthonormal basis in the orthogonal complement space of @math .
@cite_13 @cite_4 @cite_12
2935225848 2940712159 2098598867
Given @math and @math we consider a family of functionals, the @math -frame potentials FP @math , defined on the set of all collections of @math unit-norm vectors in @math . For the special case @math and @math , both the minima and the minimizers of these potentials have been thoroughly investigated. In this paper, we investigate the minimizers of the functionals FP @math , by first establishing some general properties of their minima. Thereafter, we focus on the special case @math , for which, surprisingly, not much is known. One of our main results establishes the unique minimizer for big enough @math . Moreover, this minimizer is universal in the sense that it minimizes a large range of energy functions that includes the @math -frame potential. We conclude the paper by reporting some numerical experiments for the case @math , @math , @math . These experiments lead to some conjectures that we pose. In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that the obtained energy bounds are sharp for several infinite families of codes. We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory. Furthermore, we consider similar optimal configurations in terms of random distribution of points on the sphere. In this probabilistic setting, we characterize these optimal distributions by means of probabilistic frames. Our work also indicates some connections between statistical shape analysis and frame theory.