In this paper we study the compact and convex sets
K ⊆ Ω ⊆ ℝ2 that minimize \begin{equation*} \int_{\Omega} \dist(\x,K) \,{\rm d}\x +
\lambda_1 {\rm Vol}(K)+\lambda_2 {\rm Per}(K) \end{equation*} for some constants λ1 and
λ2, that could possibly be zero. We compute in particular
the second order derivative of the functional and use it to exclude smooth points of
positive curvature for the problem with volume constraint. The problem with perimeter
constraint behaves differently since polygons are never minimizers. Finally using a purely
geometrical argument from Tilli [J. Convex Anal. 17 (2010)
583–595] we can prove that any arbitrary convex set can be a minimizer when both perimeter
and volume constraints are considered.