On a 1-capacitary type problem in the plane

We study a $1$-capacitary type problem in $\R^2$: given a set $E$, we minimize the perimeter (in the sense of De Giorgi) among all the sets containing $E$ (modulo $\Huno$) and satisfying an indecomposability constraint (according to the definition by Ambrosio-Caselles-Masnou-Morel). By suitably choosing the representant of the relevant set $E$, we show that a convexification process characterizes the minimizers. As a consequence of our result we determine the $1$-capacity of (a suitable representant of) sets with finite perimeter in the plane.