  We study the properties of a set of vectors called tight frames that obtained
as the orthogonal projection of some orthonormal basis of $\R^n$ onto $\R^k.$
We show that a set of vectors is a tight frame if and only if the set of all
cross products of these vectors is a tight frame. We reformulate a range of
problems on the volume of projections (or sections) of regular polytopes in
terms of tight frames and write a first-order necessary maximum condition of
these problems. As applications, we prove new results for the problem of the
maximization of the volume of zonotopes.
