ABSTRACT. In a recent paper by Bauschke, Borwein and Li, it is discussed that various topics in convex optimization (including error bounds and constraint qualification for a system of convex inequalities, convergence rate analysis of certain projection methods, and constrained interpolation from a convex subset/cone) are closely related to one of the following three properties for two (or more) closed convex sets/cones C1 and C2 in Euclidean space and their intersection C:
P1. For points x in any bounded set, the distance from x to C is at most a constant multiple of the maximum of the distance from x to C1 and to C2.
P2. The normal cone of C equals the sum of the normal cones of C1 and C2 everywhere on C.
P3. The tangent cone of C equals the intersection of the tangent cones of C1 and C2 everywhere on C.
In the same paper, it was proven that P1 ==> P2 ==> P3 and, while P3 was shown to imply P1 in a variety of circumstances, it was left open whether the converse implications are always true. In this talk, we show that the converse implications are not always true, alas. We also discuss additional conditions under which the converse implications would be true.