Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\cv{\cup_{k=1}^mW_k}$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient condition: the boundary $\bdS$ is positively curved, and necessary condition: $\bdS$ has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasi-concavity of defining polynomials on those points on $\bdS$ where they vanish. The gaps between them are $\bdS$ having positive versus nonnegative curvature and smooth versus nonsmooth points. A sufficient condition bypassing the gaps is when some defining polynomials of $S$ are sos-concave. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set $T$, we find that the critical object is $\pt_cT$, the maximum subset of $\pt T$ contained in $\pt \cv{T}$. We prove sufficient conditions for SDP representability: $\pt_cT$ is positively curved, and necessary conditions: $\pt_cT$ has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between them are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed.