Tight frames and related geometric problems

TL;DR

This paper explores the properties of tight frames, their geometric implications, and applies these insights to optimize the volume of zonotopes, connecting frame theory with polytope volume problems.

Contribution

It establishes a new characterization of tight frames via cross products and reformulates volume maximization problems in terms of tight frames.

Findings

Tight frames are characterized by their cross products also forming tight frames.

Reformulation of volume problems of polytopes using tight frames.

New results on maximizing the volume of zonotopes.

Abstract

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.