
Journal of Convex Analysis 17 (2010), No. 2, 535551 Copyright Heldermann Verlag 2010 LMI Representations of the Convex Hulls of Quadratic Basic Semialgebraic Sets Ugur Yildiran Dept. of Systems Engineering, Yeditepe University, Istanbul, Turkey uyildiran@yeditepe.edu.tr I. Emre Kose Dept. of Mechanical Engineering, Bogazici University, Istanbul, Turkey koseemre@boun.edu.tr [Abstractpdf] \newcommand{\closure}[1]{\overline{#1}} \newcommand{\chull}[1]{\text{{\bf co}}(#1)} \newcommand{\real}{{\mathbb R}} \newcommand{\set}[1]{{\cal #1}} We are motivated by the question of when a convex semialgebraic set in $\real^n$ is equal to the feasible set of a linear matrix inequality (LMI). Given a basic semialgebraic set, $\set{V}$, which is defined by quadratic polynomials, we restrict our attention to closure of its convex hull, namely $\closure{\chull{\set V}}$. Our main result is that $\closure{\chull{\set V}}$ is equal to the intersection of a finite number of LMI sets and the halfspaces supporting $\set V$ along a particular subset of the boundary of $\set V$. As a corollary, we show that in $\real^2$, the halfspaces of concern are finite in number, so that an LMI representation for $\closure{\chull{\set V}}$ always exists. [ Fulltextpdf (247 KB)] for subscribers only. 