On Godbersen's Conjecture

We provide a natural generalization of a geometric conjecture of F'{a}ry and R'{e}dei regarding the volume of the convex hull of $K\subset {\mathbb{R}}^n$, and its negative image $-K$. We show that it implies Godbersen’s conjecture regarding the mixed volumes of the convex bodies $K$ and $-K$. We then use the same type of reasoning to produce the currently best known upper bound for the mixed volumes $V(K[j],-K[n-j])$, which is not far from Godbersen’s conjectured bound. To this end we prove a certain functional inequality generalizing Colesanti’s difference function inequality.