Abstract: Let $\Omega$ be an unbounded domain in $\mathbb{R}\times\mathbb{R}^{d}.$ A positive harmonic function $u$ on $\Omega$ that vanishes on the boundary of $\Omega$ is called a Martin function. In this talk, we show that, when $\Omega$ is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition $\Omega$ has certain symmetry with respect to the $t$-axis, and $\partial\Omega$ is sufficiently flat, then the maximum of any Martin function along a slice $\Omega\cap (\{t\}\times\mathbb{R}^d)$ is attained at $(t,0).$ This talk is based on joint work with A.-K. Gallagher and J. Lebl.
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