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In this paper we study the fusion frame potential, that is a generalizationof the Benedetto-Fickus (vectorial) frame potential to the finite-dimensionalfusion frame setting. The structure of local and global minimizers of thispotential is studied, when we restrict the frame potential to suitable sets offusion frames. These minimizers are related to tight fusion frames as in theclassical vector frame case. Still, tight fusion frames are not as frequent astight frames; indeed we show that there are choices of parameters involved infusion frames for which no tight fusion frame can exist. Thus, we exhibitnecessary and sufficient conditions for the existence of tight fusion frameswith prescribed parameters, involving the so-called Horn-Klyachko'scompatibility inequalities. The second part of the work is devoted to the studyof the minimization of the fusion frame potential on a fixed sequence ofsubspaces, varying the sequence of weights. We related this problem to theindex of the Hadamard product by positive matrices and use it to give differentcharacterizations of these minima.

The structure of minimizers of the frame potential on fusion frames