Convex Geometry of Orbits

TL;DR

This paper explores the geometric properties of convex bodies formed by group orbits, providing new insights and estimates for their inscribed and circumscribed balls, with applications in combinatorics, algebraic geometry, and calibrated geometries.

Contribution

It introduces a new simple description of the largest volume ellipsoid inside the polar of orbit convex bodies and applies it to various important examples.

Findings

Computed the radius of the largest inscribed ball in the Traveling Salesman Polytope

Estimated the radius of the Euclidean ball containing the unit comass ball

Reviewed recent results on non-negative polynomials and their geometric properties

Abstract

We study metric properties of convex bodies B and their polars B^o, 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 non-negative 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 non-negative 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^o.