  This paper presents various worst-case results on the positive semidefinite
(psd) rank of a nonnegative matrix, primarily in the context of polytopes. We
prove that the psd rank of a generic n-dimensional polytope with v vertices is
at least (nv)^(1/4) improving on previous lower bounds. For polygons with v
vertices, we show that psd rank cannot exceed 4ceil(v/6) which in turn shows
that the psd rank of a p by q matrix of rank three is at most
4ceil(min{p,q}/6). In general, a nonnegative matrix of rank (k+1 choose 2) has
psd rank at least k and we pose the problem of deciding whether the psd rank is
exactly k. Using geometry and bounds on quantifier elimination, we show that
this decision can be made in polynomial time when k is fixed.
