KTH/SU Mathematics Colloquium
Alexander Barvinok, Univ. of Michigan at Ann Arbor
Convex geometry of orbits
Suppose that a compact group acts in a finite-dimensional real vector
space, we
pick a point in the space and consider the convex hull of its orbit. A
number of
objects of a general mathematical interest appear this way, or as polar
duals
to such convex hulls. Examples include the set of everywhere
non-negative
polynomials, the convex hull of the Grassmannian in the theory of
calibrated
geometries and some polytopes of interest in combinatorial optimization.
Although such convex bodies have a quite complicated combinatorial
structure, a lot can be said about their metric properties. The key is
a simple
description of the minimum volume ellipsoid containing the orbit.
The talk is based on a joint work with Grigoriy Blekherman.