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LetX be a convex domain in C n and letE be a convex subset ofX. The relative extremal functionuE,X forE inX is the supremum of the class of plurisub- harmonic functionsv 0 onX withv 1 onE. We show that ifE is either open or compact, then the sublevel sets ofuE,X are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

Convexity of sublevel sets of plurisubharmonic extremal functions