# Convexity of level lines of Martin functions and applications

Source: [Convexity of level lines of Martin functions and applications](<https://www.sota2.com/research/paper/68fa56381e209131ce99aae9>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:55:50.867Z

Authors: A.-K. Gallagher, J. Lebl, K. Ramachandran

Published: 2017-10-01T02:54:07.000Z

## 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 $\\Omega$ is symmetric, then the maximum of any Martin function along a slice $\\Omega\\cap (\\{t\\}\\times\\mathbb{R}^d)$ is attained at $(t,0).$

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1710.00280>)

[PDF](<https://arxiv.org/pdf/1710.00280.pdf>)