Translation surfaces with no convex presentation

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).