The concentration of measure phenomenon is one of the most fundamental tools in modern probability theory. It states that functions which are sufficiently smooth are almost constant with overwhelming probability. This probabilistic principle has found various applications in Geometric Functional Analysis, in particular in finding local, almost Euclidean structure in high-dimensional normed spaces.
In the first part of the talk, I will review the classical concentration in Gauss’ space and discuss its inadequacy for providing optimal results in several problems. This leads to the notion of super-concentration. In the second part of the talk, I will present a new variance-sensitive concentration inequality, which exploits the convexity properties of Gaussian measure. I will explain how this is connected to super-concentration and local unconditional structure. This perspective leads to new results and improvements on the best-known estimates in some long-standing open problems in high-dimensional convex geometry.