Log-Concavity of Conic Intrinsic Volumes

Let $n\ge 1$ and $C\subseteq {\mathbb{R}}^n$ be a closed convex cone with conic intrinsic volumes $v_0(C),\dots ,v_n(C)$. We prove the long-standing log-concavity conjecture for this sequence, in the stronger form $$v_k(C{)}^2\ge {\rho }_k{\rho }_{n-k}{v}_{k-1}(C)v_{k+1}(C),1\le k\le n-1,$$ where, for $l\ge 1$, ${\rho }_l=\frac{l+1}{l}\frac{{\omega }_{l-1}{\omega }_{l+1}}{{\omega }_l^2}>1$ and ${\omega }_j$ is the volume of the Euclidean unit ball in ${\mathbb{R}}^j$. The proof applies the Alexandrov–Fenchel inequality to the Takemura–Kuriki identity $$V(A[k],D[n-k])=\frac{{\omega }_k{\omega }_{n-k}}{\left(\genfrac{}{}{0px}{}{n}{k}\right)}{v}_k(C),A=C\cap B^n,D=C^{\circ }\cap B^n,$$ where $C^{\circ }$ is the polar cone, $B^n$ 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.