发表机构
McMaster University; IMDEA Software Institute; Tepper School of Business, Carnegie Mellon University; The Voleon Group（麦克马斯特大学; IMDEA软件研究所; 卡内基梅隆大学泰珀商学院; Voleon集团）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明多面体的面距离与直径之比乘以维数平方根由正则单纯形唯一最大化，验证了Peña-Wirth猜想并确定最小常数为√2，同时计算了Birkhoff多面体的面距离。
AI 中文摘要
多面体的面距离（等价于其金字塔宽度）是一种几何条件度量，出现在Frank-Wolfe算法的收敛界中。Peña和Wirth为该量建立了微积分规则，并猜想面距离与直径之比乘以维数的平方根受常数限制。我们证明了该量在每个维度中由正则单纯形唯一最大化，这意味着该猜想成立，且√2是最小常数。我们还计算了Birkhoff多面体的面距离。
英文摘要
The facial distance of a polytope, equivalently its pyramidal width, is a geometric condition measure that appears in the convergence bounds for Frank--Wolfe algorithms. Peña and Wirth developed calculus rules for this quantity and conjectured that the ratio of the facial distance to the diameter multiplied by the square root of the dimension is bounded by a constant. We show that this quantity is uniquely maximized by a regular simplex in each dimension, which implies that the conjecture holds and that $\sqrt{2}$ is the smallest constant. We also compute the facial distance of the Birkhoff polytope.
Comments9 pages
