# Convexity of level lines of Martin functions and applications

## 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 note, 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).

## Notes

### Acknowledgements

We thank Alexandre Eremenko for useful discussions and suggestions. We are also greatly indebted to the referee for pointing out an error in an earlier version of Theorem 1.2.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no competing interests.

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