Some geometric results in semidefinite programming

The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semideenite Programs, called here Spectrahedra. We rst develop a characterization for the faces of spectrahedra. More speciically, given a point x in a spectrahedron, we derive an expression for the minimal face containing x. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.