Log-Concavity of Conic Intrinsic Volumes

Abstract

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 \[ v_k(C)^2\ge ρ_kρ_{n-k}v_{k-1}(C)v_{k+1}(C), \qquad 1\le k\le n-1, \] 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 \[ V(A[k],D[n-k])=\frac{ω_kω_{n-k}}{\binom nk}v_k(C),\qquad A=C\cap B^n,\quad D=C^\circ\cap B^n, \] 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.

6 pages