The stadium does not minimize lambda(2) amongst convex plane domains

In this Note, we are interested in the minimization of the second eigenvalu e of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the "s tadium", convex hull of two identical tangent disks (Troesch conjecture). W e refute this conjecture. Nevertheless, we prove that the optimum has two p arallel segments in its boundary. (C) 2001 Academie des sciences/Editions s cientifiques et medicales Elsevier SAS.