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I will discuss a sharp estimate on the log-concavity of the first eigenvalue for the Dirichlet Laplacian (or Laplacian with potential) on a convex Euclidean domain,and the application of this estimate to prove a sharp lower bound on the fundamental gap $lambda_2-lambda_1$ in terms of the diameter of the domain.Some recent extensions of the ideas to other operators will also be discussed.This is joint work with Julie Clutterbuck and David Shellard.