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.