Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Monge–Ampère of Pac-Man

  • 31 Accesses


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.

This is a preview of subscription content, log in to check access.

Fig. 1


  1. 1.

    Baran, M.: Complex equilibrium measure and Bernstein type theorems for compact sets in \({\mathbb{R}}^{n}\). Proc. AMS 123(2), 485–494 (1995)

  2. 2.

    Bedford, E., Taylor, B.A.: The complex equilibrium measure of a symmetric convex set in \({\mathbb{R}}^{n}\). Trans. AMS 294(2), 705–717 (1986)

  3. 3.

    Bayraktar, T., Bloom, T., Levenberg, N.: Pluripotential theory and convex bodies. Mat. Sb. 209(3), 352–384 (2018)

  4. 4.

    Klimek, M.: Pluripotential Theory. Oxford University Press, Oxford (1991)

  5. 5.

    Sigurdsson, R., Snaebjarnarson, A.S.: Monge–Ampère measures of plurisubharmonic exhaustions associated to the Lie norm of holomorphic maps. Ann. Pol. Math. 123, 481–504 (2019)

Download references

Author information

Correspondence to Norm Levenberg.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by Simons Foundation Grant No. 354549.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Levenberg, N., Ma’u, S. Monge–Ampère of Pac-Man. Arch. Math. 114, 343–352 (2020).

Download citation


  • Monge–Ampère
  • Pac-Man
  • Extremal function

Mathematics Subject Classification

  • 32U15