Log-Brunn–Minkowski inequality

Prove or refute the log-Brunn–Minkowski inequality, asserting that for origin-symmetric convex bodies K and L in R^n and for λ in [0,1], the volume is log-convex under the logarithmic (p = 0) Minkowski combination; equivalently, if the support function of the 0-sum satisfies h_{(1−λ)⋅K ⊕_0 λ⋅L}(u) = h_K(u)^{1−λ} h_L(u)^λ for u in S^{n−1}, then V((1−λ)⋅K ⊕_0 λ⋅L) ≥ V(K)^{1−λ} V(L)^λ.

Background

The authors note that the uniqueness aspect of the logarithmic Minkowski problem is closely related to the log-Brunn–Minkowski inequality, a central conjecture in convex geometry which strengthens classical Brunn–Minkowski-type statements in the log setting.

This conjecture has motivated much recent work and is a focal point linking functional inequalities with geometric measure prescriptions such as cone-volume measure.