Convex hulls of curves in $n$-space

TL;DR

This paper investigates the semidefinite extension degree of convex hulls of one-dimensional semialgebraic sets in n-space, establishing a tight upper bound that measures the complexity of semidefinite representations.

Contribution

The paper proves a new upper bound for the semidefinite extension degree of convex hulls of 1D semialgebraic sets in n-space, extending known results to higher dimensions.

Findings

Bound ${ m sxdeg}(K) \\le 1 + \\lfloor n/2 \\rfloor$ for convex hulls of 1D semialgebraic sets.

The bound is tight in several cases, indicating optimality.

The result generalizes previous findings for n=2 and monomial curves.

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 ${\mathrm{sxdeg}}(K)\le1+\lfloor\frac n2\rfloor$. 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.