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

详细信息

发表期刊/会议

Transactions of the American Mathematical Society

发表日期

1966-01-01

发表年份

1966

摘要

i) 0. Summary.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).Two more proofs of Theorem 4.1 are given, one using support functions (in 6), the other using support functions, level sets and polars (in 7).Theorems 5.1 and 6.3 together show that X"-*X if and only if h(X")->int h(X).Theorem 6.1 states that [Xn,f\-*[X,f~\ if and only if/" - inf /, and Theorem 6.2 that/n - "f / if and only if </>" - nf 0, where (j> is the conjugate function of /, <>" of /".In Theorem 7.1 it is shown that /" - nf / implies La(f") -* La(f) provided a ^ inf'/.Theorem 7.2 shows that Xn-> X implies X*-+X*.Several examples are given to show that the theorems are false without the various conditions made, such as convexity.Examples are also given to show that in general pointwise and infimal convergence of functions are incomparable.1. Introduction.The present study on sequences of convex sets was motivated by the following question that arose in a proof of the optimum property of sequential probability ratio tests [1]: Given a closed, convex set X and a sequence of closed, convex sets Xn in the plane; let x be a point that is in none of the Xn, nor in X, and consider the two supporting lines through x of Xn, and of X : then if X" converges to X in some reasonable sense (to be made precise later), is the same true for the corresponding supporting lines [1, 4 and 5]?This question can