A nonlinear extension of Hoffman's error bounds for linear inequalities

In a recent paper Li and Singer introduced the notion of global error bound for a convex multifunction at a point of its domain. They showed the existence of such a global error bound when the image of the multifunction at the respective point is bounded and conjectured a result for the case when the image is not bounded. In this paper we solve their conjecture with a positive answer. For this we establish a criterion for the existence of a global error bound using the Pompeiu–Hausdorff excess. We also improve slightly some results of Li and Singer and introduce a gage associated to a multifunction similar to that for well-conditioning of convex functions, with similar properties.