Leírás
Absztrakt:
We will discuss the properties of a set of vectors called tight frames
that obtained as the orthogonal projection of some orthonormal basis of
$\R^n$ onto $\R^k$.
The tight frames appear and are used naturally in different branches of
mathematics: from quantum mechanics and approximation theory to classical
problems of convex analysis, starting with the well-known John condition
for the ellipsoid of maximal volume in a convex body.
We discuss how to write a first-order approximation formula for a
perturbation of a tight frame and use it to get some new necessary
conditions for extrema in different problems, such as the minimal and
maximal volume of projections of cross-polytope onto a subspace of a
fixed dimension, some bounds on the volume of zonotopes and etc.
Particularly, we show that the standard octahedron has the biggest volume
among all the projections of the $n$-dimensional cross-polytope onto
3-dimensional subspaces.
Based on papers:
G. Ivanov, Tight frames and related geometric problems, arXiv:1804.10055
G. Ivanov, On the volume of projections of cross-polytope, arXiv:1808.09165