Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Abstract

We show that the closed convex hull of any one-dimensional semialgebraic subset of $\mathbb{R}^n$ is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve $C$ and a compact semialgebraic subset $K$ of its $\mathbb{R}$-points, the preordering $\mathscr{P}(K)$ of all regular functions on $C$ that are nonnegative on $K$ is known to be finitely generated. Our main result, from which all others are derived, says that $\mathscr{P}(K)$ is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of $\mathscr{P}(K)$. We also extend this last result to the case where $K$ is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we show that every convex semialgebraic subset of $\mathbb{R}^2$ is a spectrahedral shadow.