We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak $$\epsilon $$ -nets.

中文翻译：

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关于弱 $$\epsilon $$-Nets 和氡数

我们表明氡数表征了可分离凸空间中弱网络的存在（欧几里得凸概念的抽象）。当氡数有限时，弱网络的构建基于 Helly 属性和 VC 类的度量属性。当氡数很大时，弱网络大小的下限取决于 Kneser 图的色数。作为应用，我们证明了弱 $$\epsilon $$ -nets 的放大结果。