发表机构
Georgia Institute of Technology; University of Waterloo（佐治亚理工学院; 滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了欧几里得球的线性矩阵不等式表示和半定扩展的紧下界，分别达到 $n$ 和 $2\sqrt{n-1}$，与现有构造匹配，并推广到极端点为半代数集的凸集。
AI 中文摘要
我们给出了欧几里得球的任何线性矩阵不等式表示大小的紧下界，以及球的半定扩展复杂性的紧下界。具体而言，我们证明任何表示欧几里得球的线性矩阵不等式必须涉及大小至少为 $n$ 的矩阵，任何欧几里得球的谱面扩展必须涉及大小至少为 $2\sqrt{n-1}$ 的矩阵。这些与现有显式构造给出的界相匹配。我们的证明依赖于关于谱面面集维数的基本事实，并且确实扩展到任何其极端点形成半代数集的凸集。
英文摘要
We present tight lower bounds for the size of any linear matrix inequality representation of the Euclidean ball, and on the semidefinite extension complexity of the ball. Specifically, we show that any linear matrix inequality representing the Euclidean ball must involve matrices of size at least $n$, and any spectrahedral extension of the Euclidean ball must involve matrices of size at least $\lceil 2 \sqrt{n-1}\rceil$. These match the upper bounds given by existing explicit constructions exactly. Our proofs rely on elementary facts about the dimension of the set of faces of spectrahedra, and indeed extend to any convex set whose extreme points form a semialgebraic set.
