Integrating Spectrahedra

TL;DR

This paper introduces the concept of the fiber body, an average of spectrahedra fibers under a linear map, and explores its geometric structure, especially for Gram spectrahedra of binary sextics and ternary quartics.

Contribution

It develops methods to analyze the boundary structure of fiber bodies and applies these to specific cases of Gram spectrahedra, revealing new geometric insights.

Findings

Fiber body of binary sextics has exactly one full-dimensional face.

Gram spectrahedron of a generic positive binary sextic has four such points.

The structure of fiber bodies changes significantly for ternary quartics.

Abstract

Given a linear map on the vector space of symmetric matrices, every fiber intersected with the set of positive semidefinite matrices is a spectrahedron. Using the notion of the fiber body we can build the average over all such fibers and thereby construct a compact, convex set, the fiber body. We show how to determine the dimensions of faces and normal cones to study the boundary structure of the fiber body given that we know about the boundary of the fibers. We use this to study the fiber body of Gram spectrahedra in the case of binary sextics and ternary quartics and find a large amount of structure on the fiber body. We prove that the fiber body in the case of binary sextics has exactly one face with a full-dimensional normal cone, whereas the Gram spectrahedron of a generic positive binary sextic has four such points. The fiber body in the case of ternary quartics changes drastically.