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arXiv · 1902.05110

Monge-Amp\`ere of Pac-Man

Norm Levenberg·Sione Ma'u
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Abstract

We show that the Monge-Amp\`ere density of the extremal function $V_P$ for a non-convex Pac-Man set $P\subset {\bf R}^2$ tends to a finite limit as we approach the vertex $p$ of $P$ linearly but with a value that may vary with the line. On the other hand, along a tangential approach to $p$ the Monge-Amp\`ere density becomes unbounded. This partially mimics the behavior of the Monge-Amp\`ere density of the union of two quarter disks set $S$ of Sigurdsson and Snaebjarnarson. We also recover their formula for $V_S$ by elementary methods.

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Norm Levenberg, Sione Ma'u. 2019-02-13. Monge-Amp\`ere of Pac-Man. https://arxiv.org/abs/1902.05110

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