A compact set admits a q-convex function in a neighborhood if and only if its q-nucleus, the maximal q-pseudoconcave subset, is empty.
V., On compact subsets possessing strictly plurisubharmonic func- tions, Izv
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On compact sets possessing $q$-convex functions
A compact set admits a q-convex function in a neighborhood if and only if its q-nucleus, the maximal q-pseudoconcave subset, is empty.