The measure concentration phenomenon is ubiquitous in local Banach geometry,probability and combinatorics. It refers to the situation where "almostall" values of a continuous function on a higher-dimensional spaceconcentrate around some most frequent value. For example, the measure concentrationon the sphere, described by classic Levy's Lemma, is a far-reaching generalizationof the fact that the volume of a higher-dimensional sphere concentratesaround its hyperplane section. In this talk I am going to present some unexpectedconsequences of the measure concentration phenomenon for several famousproblems in optimization.