The volume of polytopes and the tight frames

The projection of an orthonormal basis of $\mathbb{R}^n$ onto $\mathbb{R}^k$ is called a tight frame. This object appears in different areas of mathematics starting with quantum mechanics and convex analysis. In this talk, we will discuss the use of tight frames in the volume extrema problems for centrally symmetric polytopes. Namely, we will consider zonotopes, sections of the $n$-cube, projections of the $n$-cross-polytope, and then we will discuss how to reformulate the volume extrema problems for these classes of polytopes in terms of tight frames. The volume extrema problems for these polytopes has been study intensively. Yet there are a lot of open questions in the area. We will show how to write a 'naive' variational principle for such problems, and then we will use it to obtain the geometric necessary conditions for extremizers and some new bounds on the volume of centrally symmetric polytopes.
Zoom-conference identificator: 967 1517 7131; Password: 016296