We show that for bounded domains in \({\mathbb {C}}^n\) with \({\mathcal {C}}^{1,1}\) smooth boundary, if there is a closed set F of \(2n-1\) -Lebesgue measure 0 such that \(\partial \Omega {\setminus } F\) is \({\mathcal {C}}^{2}\) -smooth and locally pseudoconvex at every point, then \(\Omega \) is globally pseudoconvex. Unlike in the globally \({\mathcal {C}}^{2}\) -smooth case, the condition “F of (relative) empty interior” is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.
