Remarks on the Gromov-Milman inequality (Q2736129)

The estimates of concentration functions and related topics are discussed. We need some definitions to formulate the main result. Let \((M,\rho)\) be a metric space endowed with a Borel probability \(\mu.\) For a function \(g(x),\) \(x\in M,\) we define the generalized gradient NEWLINE\[NEWLINE |\nabla g(x)|=3D\lim_{\rho(x,y)\rightarrow 0}\frac{|g(x)-g(y)|}{\rho(x,y)},\quad x\in M, NEWLINE\]NEWLINE where \(|\nabla g(x)|=3D0\) if \(x\) is an isolated point. The generalized gradient exists for every locally-Lipshits function. By the definition the triple \((M,\rho,\mu)\) satisfies the Poincaré type inequality if there exists a positive constant \(\lambda\) such that NEWLINE\[NEWLINE \lambda E(g-Eg)^2\leq E|\nabla (g)|^2 NEWLINE\]NEWLINE for every locally-Lipshits function \(g(x),\) \(E|\nabla g|^2<\infty.\) \(E\) is the sign of mathematical expectation with respect to the probability \(\mu.\) The main result is the following theorem: If the triple satisfies the Poincaré type inequality, then NEWLINE\[NEWLINE \left (1-\mu(A^h)\right)\mu(A)\leq 9e^{-2h\sqrt{\lambda}} NEWLINE\]NEWLINE for every \(h>0\) and every Borel set \(A\subseteq M,\) where \(A^h\) is the \(h\)-neighbourhood of \(A.\) The main result and some other related results can be generalized for products of finite copies of \(M\) endowed with an appropriate metric.