2026/07/19 by Houshan Fu, Suijie Wang
#math.CO
Let n≥1 and C⊆ℝn be a closed convex cone with conic intrinsic volumes v0(C),…,vn(C). We prove the long-standing log-concavity conjecture for this sequence, in the stronger form vk(C)2≥ ρkρn-kvk-1(C)vk+1(C), 1≤ k≤ n-1, where, for l≥1, ρl=(l+1)/(l)\fracωl-1ωl+1ωl2>1 and ωj is the volume of the Euclidean unit ball in ℝj. The proof applies the Alexandrov--Fenchel inequality to the Takemura--Kuriki identity V(A[k],D[n-k])=\fracωkωn-k\binom nkvk(C), A=C∩ Bn, D=C^∘∩ Bn, where C^∘ is the polar cone, Bn is the Euclidean unit ball, and repeated arguments are indicated by brackets. A Master Steiner argument gives the identity directly for arbitrary closed convex cones, including degenerate ones. We also give a circular-cone counterexample to the standard ultra-log-concavity normalizations.