Convex Geometry of Orbits

Summary

We study metric properties of convex bodies B and their polars B°, where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G = S_n, the symmetric group), the set of nonnegative polynomials in real algebraic geometry (G = SO(n), the special orthogonal group), and the convex hull of the Grassmannian and the unit comass ball in the theory of calibrated geometries (G = SO(n), but with a different action). We compute the radius of the largest ball contained in the symmetric Traveling Salesman Polytope, give a reasonably tight estimate for the radius of the Euclidean ball containing the unit comass ball and review (sometimes with simpler and unified proofs) recent results on the structure of the set of nonnegative polynomials (the radius of the inscribed ball, volume estimates, and relations to the sums of squares). Our main tool is a new simple description of the ellipsoid of the largest volume contained in B°.

1. Introduction and Examples

Let Gbea compact group acting in a finite-dimensional real vector space V and let υ ∈ V be a point. The main object of this paper is the convex hull

B = B( υ) = conv(g υ : g ∈ G)

Objects such as B and B° appear in many different contexts. We give three examples below.