  We establish lower bounds on the volume and the surface area of a geometric
body using the size of its slices along different directions. In the first part
of the paper, we derive volume bounds for convex bodies using generalized
subadditivity properties of entropy combined with entropy bounds for
log-concave random variables. In the second part, we investigate a new notion
of Fisher information which we call the $L_1$-Fisher information, and show that
certain superadditivity properties of the $L_1$-Fisher information lead to
lower bounds for the surface areas of polyconvex sets in terms of its slices.
