Shadow systems and volume of polar convex bodies
  We prove that the reciprocal of the volume of the polar bodies, about the Santal\'o point, of a {\em shadow system} of convex bodies $K_t$, is a convex function of $t$. Thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of $t$ is also investigated and is characterized under certain conditions.   We apply these results to prove exact reverse Santal\'o inequality for polytopes in $\rd{d}$ that have at most $d+3$ vertices. 
Mathieu Meyer and Shlomo Reisner