For every integer n≥ 2, every pair of convex bodies K,L⊂R^n containing the origin, and every t∈[0,1], prove that V big((1-t)· K+_0t· L big)≥ V(K)^1-tV(L)^t, where the logarithmic Minkowski combination is defined by h_(1-t)· K+_0t· L(u)=h_K(u)^1-th_L(u)^t for all u∈S^n-1.

Dar’s conjecture and the log-Brunn–Minkowski inequality

The problem concerns Dar’s conjecture and the log-Brunn–Minkowski inequality. In September 2026, a preprint claimed a common proof in the plane and proposed, but did not prove, an extension to higher dimensions.

Known results

Xi and Leng proved Dar’s conjecture and established its equivalence with the log-Minkowski inequality for non-symmetric convex bodies.
The log-Brunn–Minkowski inequality was previously known in the plane.
The inequality was known for unconditional convex bodies in $\mathbb{R}^n$ .
A local inequality in dimension $2$ for $p=0$ gave another planar proof.

September 2026 unified planar proof claim

An arXiv preprint claims a unified “reverse-to-forward” argument proving both conjectures in dimension $2$ , and presents only a proposed strategy in higher dimensions. The proof is unrefereed and has not been independently verified in the retrieved sources.

Current status (as of September 2026): The planar cases are claimed proved but remain unverified; the higher-dimensional extension remains open.

Dar’s conjecture and the log-Brunn–Minkowski inequality

Primary source

Additional references

Progress summary

Solutions 0