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Convexity of sublevel sets of plurisubharmonic extremal functions

Finnur LárussonPatrice LassereRagnar Sigurdsson — 1998

Annales Polonici Mathematici

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.

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