2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164766We prove that the reciprocal of the volume of the polar bodies, about the Santaló 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 Santaló inequality for polytopes in $\rd{d}$ that have at most $d+3$ vertices.Mathematika 53 (2006), 129-148, here are some corrections of misprintsMetric GeometryFunctional Analysistext