n recent years, it was proven that there exist precisely four order isomorphisms acting in the class of geometric convex functions. These are the Legendre transform L, the geometric duality transform A, their composition J, and the identity. It is known that L and A satisfy Santal\'{o}-type inequalities, e.g. the quantity M(f)=Vol(f)⋅Vol(Lf) is bounded from above and below (here Vol(f) stands for the integral over Rn of e−f). We prove similar (asymptotically sharp) bounds for the quantity MJ(f)=Vol(Jf)/Vol(f), and describe the extremal functions.