Godbersen mixed-volume inequality

Prove that for every dimension n, every convex body K in R^n, and every integer j with 0≤j≤n, the mixed-volume inequality V(-K[j],K[n-j])≤{n\choose j}|K| holds, with equality only for simplices when 0<j<n and K is full-dimensional.

Background

The classical Rogers–Shephard inequality bounds the volume of the difference body DK=-K+K by {2n\choose n}|K|, with simplices as the unique full-dimensional equality cases. Expanding both sides by multilinearity suggests the stronger term-by-term inequality stated here.

The paper notes that the conjecture was historically open and had previously been known only in special cases, although the supplied manuscript also reports a recent proof for general convex bodies and equality characterization for convex polytopes. It is nevertheless explicitly formulated as a conjecture in the paper.

References

Godbersen cite{Godbersen} and later independently Makai Jr. cite{Makai} conjectured that the inequality holds terms by term:

— Around higher-order Godbersen conjectures  (2609.08612 - Fryš et al., 8 Sep 2026) in Conjecture (Godbersen conjecture), Section 1, Introduction

Expanding the left-hand side of eq:Schneider in terms of mixed volumes and using an appropriate combinatorial identity for ${np+n \choose n}$, one may again conjecture that the inequality holds term by term.

— Around higher-order Godbersen conjectures  (2609.08612 - Fryš et al., 8 Sep 2026) in Conjecture cited as [K25, Conjecture 1.2], Section 1, Introduction

Conjecture \ref{KK2} was in fact already made by Schneider in 2000, see p. 537.

— Around higher-order Godbersen conjectures  (2609.08612 - Fryš et al., 8 Sep 2026) in Conjecture cited as [K25, Conjecture 1.3], Section 1, Introduction

Motivated by Conjecture 2, we conjecture the corresponding unbalanced Schneider--Rogers--Shephard type inequality, analogous to Conjecture \ref{COn}:

— Around higher-order Godbersen conjectures  (2609.08612 - Fryš et al., 8 Sep 2026) in Conjecture \ref{COn2}, Section 5, subsection “Higher-order unbalanced difference body II”