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.
An extension is given of a recent result of Glazyrin, showing that an orthonormal basis @math joined with the vectors @math , where @math minimizes the @math -frame potential for @math over all collections of @math vectors @math in @math . Frames are interesting because they provide decompositions in applications where bases could be a liability. Tight frames are valuable to ensure fast convergence of such decompositions. Normalized frames guarantee control of the frame elements. Finite frames avoid the subtle and omnipresent approximation problems associated with the truncation of infinite frames. In this paper the theory of finite normalized tight frames (FNTFs) is developed. The main theorem is the characterization of all FNTFs in terms of the minima of a potential energy function, which was designed to measure the total orthogonality of a Bessel sequence. Examples of FNTFs abound, e.g., in R3 the vertices of the Platonic solids and of a soccer ball are FNTFs. For a set of @math unit vectors @math in @math , by a @math -frame potential we mean @math . In this note, we connect the minimization problem of the @math -frame potential to a certain optimization problem for real functions and find new lower bounds for some values of parameters, particularly, when @math . 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.