Sharp Lp Rogers–Shephard inequality in dimensions n ≥ 3

Prove the sharp Lp Rogers–Shephard inequality for general convex bodies in R^n with n ≥ 3 by determining the optimal constant C_{n,p} such that, for every convex body K ⊂ R^n and every p ≥ 1 (with 1/p + 1/q = 1), the inequality |K ⊕_p (−K)| ≤ C_{n,p} |K| holds. Establish the exact value of C_{n,p} and identify whether extremal bodies exist and, if so, characterize them.

Background

The classical Rogers–Shephard inequality provides a sharp upper bound on the volume of the difference body K − K of a convex body K. Bianchini and Colesanti proved a sharp analog for the p-difference body in the plane (n = 2), extending the inequality to Firey Lp sums.

In this paper, the authors prove a sharp Lp-type Rogers–Shephard inequality for the class of locally anti-blocking bodies, giving an explicit optimal constant κ_{n,q}. However, the general sharp inequality for arbitrary convex bodies in higher dimensions remains unresolved.

References

The p sum analog of this inequality was proved in R by Bianchini and Colesanti [8] and it is an open question to prove the sharp L -Rogers-Shephard inequality in R , n ≥ 3.

— On the volume of sums of anti-blocking bodies  (2409.14214 - Manui et al., 2024) in Introduction (paragraph discussing Rogers–Shephard and Lp sums)

In this connection, motivated by Conjecture 2, we propose

— Around higher-order Godbersen conjectures  (2609.08612 - Fryš et al., 8 Sep 2026) in Conjecture \ref{COn}, Section 1, subsection “Higher-order unbalanced difference bodies and weighted inequalities”

Finally, we propose the following unbalanced Fáry--Rédei type conjecture, which generalizes Conjecture 1.2:

— Around higher-order Godbersen conjectures  (2609.08612 - Fryš et al., 8 Sep 2026) in Conjecture \ref{QQ}, Section 1, subsection “Inequalities for higher-order unbalanced joins of convex bodies”