We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x + y = p lying in the positive quadrant. Every coloopless paving matroid is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases. For this latter class we give a proof that TM (a, a) ≤ max{TM (2a, 0), TM (0, 2a)} for a ≥ 2. We co...