发表机构
Institute of Mathematics, University of Warsaw（华沙大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为无条件对数凹向量提出 Sudakov 极小化原理的证明候选，通过稀疏支撑约化、阈值见证选取与指数矩估计等步骤，给出两个关于覆盖数和集中性的推论。
AI 中文摘要
Sudakov 极小化问题询问：一族彼此分离的随机线性形式是否必然具有大的期望最大值。我们为所有无条件对数凹向量提出了该原理的一个证明候选方案。论证从经典的稀疏坐标支撑约化开始，然后为每个标签选取一个可行的阈值见证，移除公共坐标核心，并控制剩余的交叉重叠。主要分析步骤是对软化见证收益的一个指数矩估计。我们通过将一维截断不等式与三角输运相结合来获得该估计，利用了源的对数凹性和目标的符号对称性。最后，一个 Bernoulli 比较恢复了符号。本文的阐述包括历史背景、证明指南、所有中间论证以及常数的显式选择。两个推论涉及矩度量下的覆盖数以及有界函数关于幅度的集中性。
英文摘要
Sudakov minoration asks whether a large family of separated random linear forms must have a large expected maximum. We present a proof candidate for this principle for all unconditional log-concave vectors. The argument starts with a classical reduction to sparse coordinate supports. It then selects one feasible threshold witness for each label, removes a common coordinate core, and controls the remaining overlaps. The main analytic step is an exponential-moment estimate for a softened witness payoff. We obtain it by combining a one-dimensional clipping inequality with triangular transport, using log-concavity of the source and sign symmetry of the target. A Bernoulli comparison restores the signs at the end. The exposition includes historical context, a guide to the proof, all intermediate arguments, and explicit choices of constants. Two consequences concern covering numbers in the moment metric and concentration of bounded functions of the magnitudes.
Comments24 pages
