Concentration Properties of Restricted Measures

We show that for any metric probability space (M, d, \mu) with a finite subgaussian constant \sigma^2(\mu) and any set A in M we have \sigma^2(\mu_A) \leq c log (e/\mu(A)) \sigma^2(\mu), where \mu_A is a restriction of \mu to the set A and c is a universal constant. We discuss examples and open problems. Based on joint work with Prasad Tetali and Sergey Bobkov.