Irregularities of distribution and geometry of planar convex sets

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When the boundary of C is C_2 we prove that for every set of N points in the unit square we have the sharp lower bound N^{1/4} for the L^2 discrepancy. When the boundary is only piecewise C_2 and is not a polygon we have the sharp lower bound N^{1/5}. We also give a whole range of intermediate sharp results between N^{1/5} and N^{1/4}. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.