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