Spectral Polyhedra
Raman Sanyal, James Saunderson
TL;DR
The paper develops a comprehensive framework for spectral convex sets $\Lambda(K)$, formed from symmetric matrices whose eigenvalues lie in a symmetric convex set $K$, and studies how these sets behave under intersections, polarity, and Minkowski sums. It establishes that $\Lambda(K)$ recovers $K$ via diagonal projection and connects to majorization and Steiner polynomials, yielding practical tools for geometric invariants. A central result is that spectral polyhedra are spectrahedra, with explicit size bounds for their representations, and smaller representations as spectrahedral shadows via Ben-Tal–Nemirovski techniques. The work further links spectral bodies to hyperbolicity cones and the generalized Lax conjecture, and explores extensions to polar representations and spectral zonotopes, outlining several open questions and directions for future research.
Abstract
A "spectral convex set" is a collection of symmetric matrices whose range of eigenvalues form a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal-Sottile-Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarity, and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies, and spectral zonotopes.