For Ω ⊂ R n , a convex and bounded domain, we study the spectrum of −∆Ω the Dirichlet Laplacian on Ω. For Λ ≥ 0 and γ ≥ 0 let ΩΛ,γ (A) denote any extremal set of the shape optimization problem, where A is an admissible family of convex domains in R n . If γ ≥ 1 and {Λj } j≥1 is a positive sequence tending to infinity we prove that {ΩΛ j ,γ (A)} j≥1 is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on A we characterize the possible limits of such subsequences as minimizers of the perimeter among domains in A of unit measure. For instance if A is the set of all convex polygons with no more than m faces, then ΩΛ,γ converges, up to rotation and translation, to the regular m-gon.2010 Mathematics Subject Classification. 35P15, 47A75, 49Q10.