Variance of Lipschitz functions and an isoperimetric problem for a class of product measures
Two theorems are proved. The first one is a generalization of a result from \textit{X. Fernique} [in: École d'été de probabilités de Saint-Flour XI-1981. Lect. Notes Math. 976, 1-74 (1983; Zbl 0507.60027)]. Denote by \({\mathcal F}\) be class of all functions satisfying the condition \[ \bigl|f(x)-f(y)\bigr|\leq d(x,y) =\sum^n_{k=1} |x_k-y_k |, \quad x=(x_1, \dots, x_n),\;y=(y_1, \dots, y_n)\in R^n. \] Theorem 1. Let \(\xi= (\xi_1, \dots, \xi_n)\) be a vector of independent random variables with finite variance \(\sigma^2_k= \text{Var} \xi_k\), \(1\leq k\leq n\). For any \(f\in{\mathcal F}\), \[ \text{Var} f(\xi)\leq \text{Var} (\xi_1+ \cdots+ \xi_n)= \sum^n_{k=1} \sigma^2_k. \] Let \(M^n(\sigma)\) be the family of all the product measures \(\mu= \mu_1 \otimes\cdots \otimes\mu_n\) on \(R^n\) with given variance \(\text{Var}(\mu)=\sigma^2\), where \[ \text{Var} (\mu)= \sum^n_{k=1} \int_{R^1} \Bigl|x-\int_{R^1} td\mu_k(t) \Bigr|^2 d \mu_k(x). \] With the help of this theorem the authors prove the following Theorem 2. For any \(h>0\), \(\sigma>0\) and \(p\in(0,1)\), \[ \inf_{\mu \in M^n (\sigma)} \inf_{A: \mu(A) \geq p} \mu(A^h) =\begin{cases} p, \quad & \text{if } h\leq\sigma/ \sqrt{p(1-p)}, \\ 1-p\sigma^2 /(ph^2- \sigma^2), \quad & \text{if } h\geq\sigma/ \sqrt {p(1-p)}, \end{cases} \] where \(A\) is a Borel set and \(A^h= \{x:d(x,a) <h\), for some \(a\in A\}\).
scientific article; zbMATH DE number 947102
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