In his 1996 paper, Talagrand highlighted that the Law of Large Numbers (LLN) for independent random variables can be viewed as a geometric property of multidimensional product spaces. This phenomenon is known as the concentration of measure. To illustrate this profound connection between geometry and probability theory, we consider a seemingly intractable geometric problem in multidimensional Euclidean space and solve it using standard probabilistic tools such as the LLN and the Central Limit Theorem (CLT).

中文翻译：

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高维空间怪异

在他 1996 年的论文中，Talagrand 强调，独立随机变量的大数定律 （LLN） 可以被视为多维乘积空间的几何特性。这种现象称为测度浓度。为了说明几何学和概率论之间的这种深刻联系，我们考虑了多维欧几里得空间中一个看似棘手的几何问题，并使用 LLN 和中心极限定理 （CLT） 等标准概率工具来解决它。