Second-order cone representation for convex subsets of the plane

Semidefinite programming (SDP) is the task of optimizing a linear function over the common solution set of finitely many linear matrix inequalities (LMIs). For the running time of SDP solvers, the maximal matrix size of these LMIs is usually more critical than their number. The semidefinite extension degree $\text{sxdeg}(K)$ of a convex set $K\subseteq\mathbb R^n$ is the smallest number $d$ such that $K$ is a linear image of a finite intersection $S_1\cap\dots\cap S_N$, where each $S_i$ is a spectrahedron defined by a linear matrix inequality of size $\le d$. Thus $\text{sxdeg}(K)$ can be seen as a measure for the complexity of performing semidefinite programs over the set $K$. We give several equivalent characterizations of $\text{sxdeg}(K)$, and use them to prove our main result: $\text{sxdeg}(K)\le2$ holds for any closed convex semialgebraic set $K\subseteq\mathbb R^2$. In other words, such $K$ can be represented using the second-order cone.