We show that the Cheeger constant of any logconcave density is at least Tr(A^2)^{-1/4} where A is its covariance matrix, i.e., n^{-1/4} for isotropic logconcave densities. This improves on known bounds for the KLS, thin-shell, concentration and Poincare constants, and gives an alternative proof of the current best bound for the slicing constant. We then show how our proof can be used to derive a nearly tight bound for the log-Sobolev constant of isotropic logconcave distributions. The talk is joint work with Yin Tat Lee (UW and MSR).