Convergence of sequences of convex sets, cones and functions. II

A definition is given of convergence of a sequence of sets to a set, written X" -> X, where X and the Xn are subsets of Euclidean m-space £"'.A new mode of convergence of a sequence of real valued functions on Em to a function is introduced, termed infimal convergence, and written f"-*-,"ff.P(X), A(X), h(X) and X* are the projecting cone, asymptotic cone, support function and polar, respectively, of X. [X, f~\ = {(x, a) : x e X, a ¡^ f(x)}, where X = {x :/(x) < 00}, and La(f) = {x :/(x) ^ a}.In the statements of the theorems below it is understood that all sets are convex, limit sets are nonempty, and all functions are convex and closed ( = lower semicontinuous).Theorem 3.2 states that if X" -> X and C is any open cone covering A(X) then there exist N and a disk D(R) = {x : | x | g R} such that I.cCU D(R) for n> N. Theorem 4.1 proves that if X" -> X and the origin of £mis not in X, then P(X") -* P(X).