On a positive semide nite

We study the convex body e L n deened by e L n := fX j X = (x ij) positive semideenite n n matrix ; x ii = 1 for all ig: Our main motivation for investigating this body comes from combinatorial optimization, namely from approximating the max-cut problem. An important property of e L n is that, due to the positive semideenite constraints, one can optimize over it in polynomial time. On the other hand, e L n still inherits the diicult structure of the underlying combinatorial problem. In particular, it is NP-hard to decide whether the optimum of the problem minTr(CX); X 2 e L n is reached in a vertex. This result follows from the complete characterization of the matrices C of the form C = bb t for some vector b, for which the optimum of the above program is reached in a vertex. We describe several geometric properties of e L n. Among other facts, we show that e L n has 2 n?1 vertices corresponding to all bipartitions of the set