We consider the uniform measure over a high-dimensional isotropic convex body. We prove that, up to logarithmic factors, the corresponding isoperimetric minimizers are ellipsoids. We thus establish a connection between two well-known conjectures regarding the uniform measure over a high dimensional convex body, namely the Thin-Shell conjecture and the conjecture by Kannan-Lovasz-Simonovits (KLS), showing that a positive answer to the former will imply a positive answer to the latter (up to a logarithmic factor). Our proof relies on the analysis of the eigenvalues of a certain random-matrix-valued stochastic process related to a convex body. We also discuss some algorithmic implications of our methods.