The average distance problem with perimeter-to-area ratio penalization

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 $p\ge 1$, $\text{la}>0$ are given parameters, the unknown $\mathrm{\Omega }$ varies among compact, convex, Hausdorff two-dimensional sets of ${\mathbb{R}}^2$, $\text{pd}\mathrm{\Omega }$ denotes the boundary of $\mathrm{\Omega }$, and $\text{dist}(x,\text{pd}\mathrm{\Omega }):={\inf }_{y\in \text{pd}\mathrm{\Omega }}\mathrm{\mid }x-y\mathrm{\mid }$. The integral term ${\int }_{\mathrm{\Omega }}{\text{dist}}^p(x,\text{pd}\mathrm{\Omega })\text{d}x$ quantifies the “easiness” for points in $\mathrm{\Omega }$ 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 $\text{E}$.