A famous conjecture made by Lord Rayleigh is the following: "The first eigenvalue of the Laplacian on an open domain of given measure with Dirichlet boundary conditions is minimum when the domain is a ball and only when it is a ball". This conjecture was proved simultaneously and independently by Faber [G. Faber, Beweiss dass unter allen homogenen Membranen von gleicher Flache und gleicher Spannung die kreisforfegige den leifsten Grundton gibt. Sitz. bayer Acad. Wiss. (1923) 169-172] and Krahn [E. Krahn, Uber eine von Rayleigh formulierte Minimaleigenschaftdes Kreises. Math. Ann. 94 (1924) 97-100.]. We shall deal with the p-Laplacian version of this theorem. Mathematics Subject Classification.