  In this paper we consider the functional \begin{equation*}
E_{p,\la}(\Omega):=\int_\Omega \dist^p(x,\pd \Omega )\d x+\la \frac{\H^1(\pd
\Omega)}{\H^2(\Omega)}. \end{equation*} Here $p\geq 1$, $\la>0$ are given
parameters, the unknown $\Omega$ varies among compact, convex, Hausdorff
two-dimensional sets of $\R^2$, $\pd \Omega$ denotes the boundary of $\Omega$,
and $\dist(x,\pd \Omega):=\inf_{y\in\pd \Omega}|x-y|$. The integral term
$\int_\Omega \dist^p(x,\pd \Omega )\d x$ quantifies the "easiness" for points
in $\Omega$ to reach the boundary, while $\frac{\H^1(\pd
\Omega)}{\H^2(\Omega)}$ is the perimeter-to-area ratio. The main aim is to
prove existence and $C^{1,1}$-regularity of minimizers of $\E$.
