Convex hulls of curves in $n$-space

Abstract

Let $K\subseteq {\mathbb{R}}^n$ be a convex semialgebraic set. The semidefinite extension degree $\mathrm{sxdeg}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of which is described by a linear matrix inequality of size $\le d$. This invariant can be considered to be a measure for the intrinsic complexity of semidefinite optimization over the set $K$. For an arbitrary semialgebraic set $S\subseteq {\mathbb{R}}^n$ of dimension one, our main result states that the closed convex hull $K$ of $S$ satisfies $\mathrm{sxdeg}(K)\le 1+\lfloor \frac{n}{2}\rfloor$. This bound is best possible in several ways. Before, the result was known for $n=2$, and also for general $n$ in the case where $S$ is a monomial curve.