凸集上的胞腔
Integer cells in convex sets
摘要：
R^n中的每个凸体K都有一个坐标投影PK，它至少包含整数格点PZ^n的vol（0.1 K）个胞元，只要这个体积至少为1。我们对闵可夫斯基定理的这个对应的证明是基于绍尔、谢拉和瓦普尼克-切尔沃年基斯的组合密度定理对Z^n的推广。这为研究凸体的截面提供了一种新的途径。特别是，基本结果的渐近凸几何，如体积比定理和米尔曼的对偶的直径承认自然版本的坐标部分。
英文摘要：
Every convex body K in R^n has a coordinate projection PK that contains at least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and Vapnik-Chervonenkis to Z^n. This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.
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