On unbalanced difference bodies and Godbersen's conjecture

The longstanding Godbersen’s conjecture states that for any convex body $K\subset {\mathbb{R}}^n$ of volume $1$ and any $j\in \{0,\dots ,n\}$, the mixed volume $V_j=V(K[j],-K[n-j])$ is bounded by $\left(\genfrac{}{}{0px}{}{n}{j}\right)$, with equality if and only if $K$ is a simplex. We demonstrate that several consequences of this conjecture are true: certain families of linear combinations of the $V_j$, arising from different geometric constructions, are bounded above by their values when one substitutes $\left(\genfrac{}{}{0px}{}{n}{j}\right)$ for $V_j$, with equality if and only if $K$ is a simplex. One of our results implies that for any $K$ of volume $1$ we have $\frac{1}{n+1}{\sum }_{j=0}^n{\left(\genfrac{}{}{0px}{}{n}{j}\right)}^{-1}{V}_j\le 1$, showing that Godbersen’s conjecture holds ‘‘on average’’ for any body. Another result generalizes the well-known Rogers-Shephard inequality for the difference body.