发表机构
Northwestern University; MIT Sloan School of Management（西北大学; 麻省理工学院斯隆管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了凸体的对称性测度决定其舍入因子，解决了Belloni和Freund2005年的猜想，给出了几乎紧的舍入因子，并针对点凸包型和半空间交型多面体分别提出了对应的正则化椭球优化方法。
AI 中文摘要
凸体 $S\subset\mathbb{R}^n$ 的对称性测度定义为：$$\operatorname{sym}(S):=\max\{\alpha\ge0:\text{ 存在 }x\in S\text{ 使得 }-\alpha(S-x)\subseteq S-x\},$$其中这样的 $x$ 被称为闵可夫斯基中心。我们证明每个凸体 $S$ 都存在一个 $\sqrt{\frac{n}{\operatorname{sym}(S)}}$ 舍入，即存在一个以原点为中心的椭球 $E$ 和一个中心 $c$，满足：$$E\subseteq S-c\subseteq\sqrt{\frac{n}{\operatorname{sym}(S)}}\\,E.$$该结果是 Belloni 和 Freund 于 2005 年提出的猜想。作为特例，当 $\operatorname{sym}(S) \ge 1/n$ 时，可得到已知的 $n$ 舍入结果；当 $\operatorname{sym}(S) = 1$ 时，可得到已知的 $\sqrt{n}$ 舍入结果。当 $S$ 是由点集凸包给出的多面体时，所需的舍入通过关于闵可夫斯基中心正则化的最小体积覆盖椭球问题得到；当 $S$ 是由半空间交集给出的多面体时，所需的舍入通过关于闵可夫斯基中心正则化的最大体积内接椭球问题得到。我们还证明因子 $\sqrt{\frac{n}{\operatorname{sym}(S)}}$ 在维度和对称性的依赖上是几乎紧的：当 $\frac{n+1}{1+\operatorname{sym}(S)}$ 为整数时，通过显式构造可证明该因子是紧的；在更一般的情况下，对于每个维度 $n$ 和每个可允许的对称性值，我们构造了一个凸体 $S$，使得其任意舍入的因子至少为 $\sqrt{\frac{2}{3}}\sqrt{\frac{n}{\operatorname{sym}(S)}}$。
英文摘要
The symmetry measure of a convex body $S\subset\mathbb{R}^n$ is given by: $\mathrm{sym}(S):=\max\{α\ge0:\text{ there exists }x\in S\text{ such that }-α(S-x)\subseteq S-x\}$, where such an $x$ is called a Minkowski center. We prove that every convex body $S$ admits a $\sqrt{\frac{n}{\mathrm{sym}(S)}}$-rounding of $S$, namely, there exists an origin-centered ellipsoid $E$ and a center $c$ such that $E\subseteq S-c\subseteq\sqrt{\frac{n}{\mathrm{sym}(S)}}\,E$. This result was conjectured in 2005 by Belloni and Freund. As special cases, this recovers an $n$-rounding of $S$ (since $\mathrm{sym}(S)\ge\frac{1}{n}$), and a $\sqrt{n}$-rounding when $\mathrm{sym}(S)=1$. In the case when $S$ is a polytope given as the convex hull of points, the desired rounding is produced by a regularized minimum-volume covering ellipsoid problem where the regularization is with respect to the Minkowski center. Similarly, when $S$ is a polytope given as the intersection of halfspaces, such a rounding is produced by a regularized maximum-volume inscribed ellipsoid problem. In both of these cases, the rounding can be computed by first solving a linear optimization problem (to compute $\mathrm{sym}(S)$ and a Minkowski center), and then solving a convex optimization problem with a logarithmic determinant objective, second-order cone constraints, and one semidefinite cone constraint. We also show that the factor $\sqrt{\frac{n}{\mathrm{sym}(S)}}$ is nearly tight in its dependence on dimension and symmetry. When $\frac{n+1}{1+\mathrm{sym}(S)}$ is an integer, we show by explicit construction that the factor $\sqrt{\frac{n}{\mathrm{sym}(S)}}$ is tight. In the more general case, for every dimension $n$ and every admissible symmetry value, we construct a polytope $S$ for which every rounding factor is at least $\sqrt{\frac{2}{3}}\sqrt{\frac{n}{\mathrm{sym}(S)}}$.
Comments24 pages, 1 figure
