The (B) conjecture for uniform measures in the plane

We prove that for any two centrally-symmetric convex shapes $K,L \subset $\mathbb{R}$^2$,thefunction$t \mapsto |e^t K \cap L|$ is log-concave. This extends a result of Cordero-Erausquin, Fradelizi and Maurey in the two dimensional case. Possible relaxations of the condition of symmetry are discussed.