圆锥内蕴体积的对数凹性
Log-Concavity of Conic Intrinsic Volumes
AI总结：
研究圆锥内蕴体积序列的对数凹性，通过应用亚历山德罗夫 - 芬切尔不等式于竹村 - 栗木恒等式进行证明，得到更强形式的对数凹性结论，并给出标准超对数凹性归一化的圆锥反例。
AI中文摘要：
设\(n\geq1\)且\(C\subseteq\mathbb{R}^n\)为具有圆锥内蕴体积\(v_0(C),\ldots,v_n(C)\)的闭凸锥。我们证明了该序列长期存在的对数凹性猜想，形式更强为\(v_k(C)^2\geq\rho_k\rho_{n - k}v_{k - 1}(C)v_{k + 1}(C)\)，\(1\leq k\leq n - 1\)。证明应用亚历山德罗夫 - 芬切尔不等式于竹村 - 栗木恒等式。还给出了一个关于标准超对数凹性归一化的圆锥反例。
英文摘要：
Let $n\ge1$ and $C\subseteq\mathbb{R}^n$ be a closed convex cone with conic intrinsic volumes $v_0(C),\ldots,v_n(C)$. 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 $l\ge1$, $ρ_l=\frac{l+1}{l}\frac{ω_{l-1}ω_{l+1}}{ω_l^2}>1$ and $ω_j$ 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.
