Convex hulls of curves in n-space

Abstract

Let $K\subseteq R^n$ be a convex semialgebraic set. The semidefinite extension degree $\mathrm{sxdeg}\left(K\right)$ 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 ≤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 $S\subseteq R^n$ of dimension one, our main result states that the closed convex hull K of S satisfies $\mathrm{sxdeg}\left(K\right)\le 1+\lfloor \frac{n}{2}\rfloor$. This bound is best possible in several ways. Before, the result was known for $n=2$, and also for general n in the case when S is a monomial curve.