各向同性位置下平均规范与平均宽度的几何界
Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position
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中文总结 AI 辅助
本文针对各向同性位置下原点对称凸体，通过两种几何方法推导平均规范与平均宽度的乘积上界，所用拉普拉斯体等价于质心体，核心依赖切片定理的小球推论。
中文摘要 AI 辅助
设 $K \subset \mathbb{R}^n$ 是原点对称凸体，假设其均匀概率测度经概率归一化后满足各向同性条件，即 $\int_K x \otimes x \\, d\mu_K(x) = \mathrm{Id}_n$。本文给出 $M(K) \leq C \frac{\log(n)}{\sqrt{n}}$ 和 $M^*(K) \leq C \sqrt{n} \\, \log(n)$ 的确定性几何证明，其中 $M(K) = \int_{\mathbb{S}^{n-1}} \\|\theta\\|_K \\, d\sigma(\theta)$，$M^*(K) = \int_{\mathbb{S}^{n-1}} h_K(\theta) \\, d\sigma(\theta)$。结合两估计可得 $M(K) M^*(K) \leq C \log^2(n)$。第一种证明使用二进质心体的二次聚合，第二种使用拉普拉斯体 $p\{\Lambda_K \leq p\}^\circ$ 的类似加权聚合，该类拉普拉斯体经 Klartag 与 E. Milman 的工作证明与质心体等价。两种情形下，在余维数为 $O(p)$ 的子空间外的每个二进尺度的曲率，均通过极小极大原理、勒让德对偶性及球面拉普拉斯算子推导出所需估计，唯一的高维输入是切片定理的无维小球推论。
英文摘要
Let $K \subset \mathbb{R}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \[ \int_K x \otimes x \, dμ_K(x) = \mathrm{Id}_n. \] We give deterministic geometric proofs of \[ M(K) \leq C \frac{\log(n)}{\sqrt{n}} \qquad \text{and} \qquad M^*(K) \leq C \sqrt{n} \, \log(n), \] where \[ M(K) = \int_{\mathbb{S}^{n-1}} \|θ\|_K \, dσ(θ), \qquad M^*(K) = \int_{\mathbb{S}^{n-1}} h_K(θ) \, dσ(θ). \] Combining both estimates yields \[ M(K) M^*(K) \leq C \log^2(n). \] The first proof uses a quadratic aggregate of dyadic centroid bodies. The second uses the analogous weighted aggregate of the Laplace bodies $p\{Λ_K \leq p\}^{\circ}$, which are equivalent to the centroid bodies by the work of Klartag and E. Milman. In both cases, curvature at each dyadic scale outside a subspace of codimension $O(p)$ leads, via the min--max principle, Legendre duality, and the spherical Laplacian, to the required estimate. The only high-dimensional input is the dimension-free small-ball consequence of the slicing theorem.
