Convexity of level lines of Martin functions and applications

Let $\mathrm{\Omega }$ be an unbounded domain in $\mathbb{R}\times {\mathbb{R}}^d\mathrm{.}$ A positive harmonic function $u$ on $\mathrm{\Omega }$ that vanishes on the boundary of $\mathrm{\Omega }$ is called a Martin function. In this note, we show that, when $\mathrm{\Omega }$ is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition $\mathrm{\Omega }$ is symmetric, then the maximum of any Martin function along a slice $\Omega\cap ({t}\times$\mathbb{R}$^d)$isattainedat$(t,0).$