  We introduce three quantities called total coherence, total volume, and
nuclear energy, and we show that equiangular Parseval frames maximize all three
of these quantities over the set of all Parseval frames. We then show that
equiangular Parseval frames also maximize the total volume and nuclear energy
over the set of equal norm frames. Along the way we derive a bound on the
smallest k-dimensional volume in an equal norm frame which is a generalization
of the Welch bound. We conclude with an extensive list of open questions.
