# Feasible space of SDP

Typically the non-empty feasible space of a SDP has some curved boundary which is why the feasible space has infinitely many extreme points. Is it ever possible to have a SDP whose non-empty feasible region is the convex hull of a finite (exponential) number of rank-$1$ matrices? If yes, how do we characterize such a SDP? What can we say about the affine hull of these extreme points? Is it somehow related to the affine space defined by the set of equality linear constraints?

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I would suggest first thinking about the corresponding questions for linear programs (LPs), which are after all a subclass of semidefinite programs (SDPs). Is there any reason to expect a "nice" characterization? Is there a simple answer to the affine hull question there? –  Noah Stein May 31 '11 at 14:53