# Monge-Amp\`ere of Pac-Man

## TL;DR

This paper investigates the behavior of the Monge-Ampère density of the extremal function for a non-convex Pac-Man set, revealing its finite limit along certain lines and unboundedness along tangential approaches, with connections to union of quarter disks.

## Contribution

It characterizes the Monge-Ampère density near the vertex of a non-convex Pac-Man set and relates it to known behaviors of union of quarter disks, providing elementary derivations.

## Key findings

- Monge-Ampère density tends to a finite limit linearly approaching the vertex.
- Density becomes unbounded along tangential approaches to the vertex.
- Recovered known formula for the extremal function of union of quarter disks.

## 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.