  We prove that if the given compact set $K$ is convex then a minimizer of the
functional $$ I(v)=\int_{B_R} |\nabla v|^p dx+\text{Per}(\{v>0\}),\,1<p<\infty,
$$ over the set $\{v\in H^1_0(B_R)|\,\, v\equiv 1\,\,\text{on}\,\, K\subset
B_R\}$ has a convex support, and as a result all its level sets are convex as
well. We derive the free boundary condition for the minimizers and prove that
the free boundary is analytic and the minimizer is unique.
