Perimeters, uniform enlargement and high dimensions (Q520689)

The paper under review investigates the isoperimetric problem in product spaces equipped with the supremum distance. In order to formulate the main result, let us recall the basic definitions and notations. Let \((X,d,\mu)\) be a metric probability space (\(\mu\) is defined on the Borel \(\sigma\)-algebra), and denote by \(A_r\) (\(r>0\)) the \(r\)-neighborhood of the Borel set \(A\), that is, \[ A_r:=\{x\in X\mid d(x,A)<r\}. \] The outer and inner boundary measures are defined by \[ \mu^+(A)=\liminf\limits_{r\to 0^+}\frac{\mu(A_r)-\mu(A)}{r}\quad\text{and}\quad\mu^-(A)=\mu^+(X\setminus A), \] respectively. The isoperimetric function denoted by \(I_{(X,d,\mu)}\) is defined as follows \[ I_{(X,d,\mu)}(p)=\inf_{A\subseteq X: \mu(A)=p}\min(\mu^+(A),\mu^-(A)),\qquad p\in[0,1]. \] For given metric spaces \((X_i,d_i,\mu_i)\) (\(i\in\{1,\dots,n\}\)) we equip the product probability space \((X_1\times\cdots\times X_n,\mu_1\otimes\cdots\otimes\mu_n\)) with the \(l_{\infty}\)-combination of the distances \[ d_{\infty}^{(n)}\big((x_1,\dots,x_n),(y_1,\dots,y_n)\big):=\max_{1\leq i\leq n}d_i(x_i,y_i). \] The authors describe several motivations of using uniform enlargement. For example, they explain the connection with the usual isoperimetric problem restricted to rectilinear sets (those sets appear when dealing with the supremum of random variables). Another important motivation is that isoperimetric inequalities for the uniform enlargement naturally appear in the theory of influences of variables. Using that \(I_{(X^{n+1},d_{\infty}^{(n+1)},\mu^{n+1})}\leq I_{(X^{n},d_{\infty}^{(n)},\mu^{n})}\) holds for all \(n\geq 1\), one can define the infinite dimensional isoperimetric profile of \((X,d,\mu)\) as the function \[ I_{\mu^{\infty}}(t):=\inf\limits_{n\geq1}I_{(X^{n},d_{\infty}^{(n)},\mu^{n})}(t),\qquad t\in[0,1]. \] The goal of the paper can be stated informally (without mentioning the required properties of \((X,d,\mu)\)) as follows: there exists a constant \(K>1\) such that \[ I_{(X,d,\mu)}\geq I_{\mu^{\infty}}\geq\frac{1}{K}I_{(X,d,\mu)} \] if and only if there exists a constant \(D>1\) such that for all \(0<s<t<1\) the following inequality holds \[ \frac{I_{(X,d,\mu)}(s)}{s\log(1/s)}\leq D\frac{I_{(X,d,\mu)}(t)}{t\log(1/t)}. \] In the last part of the paper, the authors obtain an extension of the Kahn-Kalai-Linial theorem about the existence of a coordinate with a large influence.