Local patching for convex sets in Cn

Abstract

We consider open sets Ω1 and Ω2 in ℂ n so that\(\bar \Omega _1 \cap \bar \Omega _2 = \{ 0\} \). We investigate the existence of a basis of pseudoconvex neighborhoods for\(\bar \Omega _1 \cup \bar \Omega _2 \) and of local (pseudoconvex or polynomially convex) “patching.” Positive results are given for strictly convex sets, and counterexamples are given for strictly pseudoconvex sets (even convex ones).