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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References
Baran, M.: Complex equilibrium measure and Bernstein type theorems for compact sets in \({\mathbb{R}}^{n}\). Proc. AMS 123(2), 485–494 (1995)
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)
Bayraktar, T., Bloom, T., Levenberg, N.: Pluripotential theory and convex bodies. Mat. Sb. 209(3), 352–384 (2018)
Klimek, M.: Pluripotential Theory. Oxford University Press, Oxford (1991)
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)
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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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DOI: https://doi.org/10.1007/s00013-019-01422-6