On compact sets possessing q-convex functions

Abstract We show that there exists a q -convex function in a neighborhood of a compact set K in a complex manifold $$\mathcal {M}$$ if and only if the q -nucleus of this compact set is empty. The latter can be characterized as the maximal q -pseudoconcave subset of K , i.e., a subset of K containing all other compact q -pseudoconcave subsets in K .