  We give infinite lists of translations surfaces with no convex presentations.
We classify the surfaces in the stratum H(2) which do not have convex
presentations, as well as those with no strictly convex presentations. We show
that in H(1,1), all surfaces in the eigenform loci E_4, E_9 or E_{16} have no
strictly convex presentation, and that the list of surfaces with no convex
presentations in H(1,1) - (E_4 union E_9 union E_{16}) is finite and consists
of square-tiled surfaces. We prove the existence of non-lattice surfaces
without strictly convex presentations in all of the strata H^{(hyp)}(g-1, g-1).
