The concentration of Lipschitz functions around their expectation is a classical topic that continues to be very active. We will discuss some recent progress, including:
1- A tight log-Sobolev inequality for isotropic logconcave densities
2- A unified and improved large deviation inequality for convex bodies
3- An extension of the above to Lipschitz functions (generalizing the Euclidean squared distance)
The main technique of proof is a simple iteration (equivalently, a Martingale process) that gradually transforms any density into one with a Gaussian factor, for which isoperimetric inequalities are considerably easier to establish. The talk is joint work with Yin Tat Lee (UW) and will involve some elementary calculus.