The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in ℝ^N. The result presented in this talk strengthens Andrews-Clutterbuck inequality by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The proof relies on a variational interpretation of the fundamental gap and the analysis of convex partitions à la Payne-Weinberger. This is a joint work with Vincenzo Amato and Ilaria Fragalà.
