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.
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.