First we recall a Faber-Krahn type inequality and an estimate for λp(Ω) in terms of the so-called Cheeger constant. Then we prove that the eigenvalue λp(Ω) converges to the Cheeger constant h(Ω) as p → 1. The associated eigenfunction up converges to the characteristic function of the Cheeger set, i.e. a subset of Ω which minimizes the ratio |∂D|/|D| among all simply connected D ⊂⊂ Ω. As a byproduct we prove that for convex Ω the Cheeger set ω is also convex.