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.

References

The solution is closely related to the conjectured log-Brunn- Minkowski inequality, a conjectured inequality that has attracted much attention (see e.g., [11,19,52,53,78]).

— Chord Measures in Integral Geometry and Their Minkowski Problems  (2502.08082 - Lutwak et al., 12 Feb 2025) in Section 1 (Introduction), Minkowski Problems

Over the past few decades, extending the strong (B)-theorem to arbitrary even log-concave measures has attracted considerable attention because of its connection with the log-Brunn--Minkowski conjecture put forth by Böröczky--Lutwak--Yang--Zhang . Indeed, it was shown by Saroglou that the log-Brunn--Minkowski conjecture is equivalent to the conjecture that \begin{equation}\label{e:LogBM} n \ni s \mapsto \gamma_{\Sigma}(\Delta(es) [-1, 1]n) \end{equation} is log-concave for any $\Sigma \in \mathcal{P}_n$ and $n \in \mathbb{N}$.

e:LogBM:

n∋s↦γΣ(Δ(es)[−1,1]n)^n \ni s \mapsto \gamma_{\Sigma}(\Delta(e^s) [-1, 1]^n)

— Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality  (2610.13596 - Milman et al., 9 Oct 2026) in Section 1, subsection “The heterovariant (B)-theorem”