Monge-Amp\`ere of Pac-Man

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.