Convexity of sublevel sets of plurisubharmonic extremal functions
Volume 68 / 1998
Annales Polonici Mathematici 68 (1998), 267-273
DOI: 10.4064/ap-68-3-267-273
Abstract
Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function $u_{E,X}$ for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of $u_{E,X}$ are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.
