# A Note on the Convex Hull of Finitely Many Projections of Spectrahedra

## TL;DR

This paper proves that the convex hull of finitely many projections of spectrahedra remains a projection of a spectrahedron, extending previous results to unbounded sets and broadening the understanding of semidefinitely representable sets.

## Contribution

It generalizes prior work by Helton and Nie, demonstrating that the convex hull of a finite union of projections of spectrahedra is also a projection of a spectrahedron, regardless of boundedness.

## Key findings

- Convex hull of finite union of spectrahedron projections is a spectrahedral projection
- Extends Helton and Nie's bounded set result to unbounded sets
- Enhances understanding of semidefinitely representable sets

## Abstract

A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of a spectrahedron. This improves upon the result of Helton and Nie, who prove the same result in the case of bounded sets.