  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.
