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Monge–Ampère of Pac-Man

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Abstract

We show that the Monge–Ampère density of the extremal function \(V_P\) for a non-convex Pac-Man set \(P\subset {{\mathbb {R}}}^2\) tends to a finite limit as we approach the vertex p of P along lines but with a value that may vary with the line. On the other hand, along a tangential approach to p, the Monge–Ampère density becomes unbounded. This partially mimics the behavior of the Monge–Ampère density of the union of two quarter disks S of Sigurdsson and Snaebjarnarson (Ann Pol Math 123:481–504, 2019). We also recover their formula for \(V_S\) by elementary methods.

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Correspondence to Norm Levenberg.

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Levenberg, N., Ma’u, S. Monge–Ampère of Pac-Man. Arch. Math. 114, 343–352 (2020). https://doi.org/10.1007/s00013-019-01422-6

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Keywords

  • Monge–Ampère
  • Pac-Man
  • Extremal function

Mathematics Subject Classification

  • 32U15