We analyze the Poincar & eacute; and Log-Sob olev constants of logconcave densities in Rn. For the Poincar & eacute; constant, we give an improved estimate of O(root n) for any isotropic logconcave density. For the Log-Sob olev constant, we prove a bound of Omega(1/D), where D is the diameter of the support of the density, and show that this is asymptotically the best possible bound, resolving a question posed by Frieze and Kannan in 1997. These bounds have several interesting consequences. Improved bounds on the thin-shell and Cheeger/KLS constants are immediate. The ball walk to sample an isotropic logconcave density in Rn converges in O*(n2.5) steps from a warm start, and the speedy version of the ball walk, studied by Kannan, Lov & aacute;sz and Simonovits mixes in O*(n2D) proper steps from any start, also a tight bound. As another consequence, we obtain a unified and improved large deviation inequality for the concentration of any L-Lipshitz function over an isotropic logconcave density (studied by many), generalizing bounds of Paouris and Guedon-E. Milman. Our proof technique is a development of stochastic localization, first introduced by Eldan.
