# Abstract Convex Analysis (Wiley-Interscience and Canadian by Ivan Singer

By Ivan Singer

This publication examines summary convex research and offers the result of contemporary examine, in particular on parametrizations of Minkowski kind dualities and of conjugations of variety Lau. It explains the most ideas via instances and distinctive proofs.

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**Example text**

125) sup w(G) (where max denotes a sup which is attained for some WO E F*). 125) may be regarded as one of strong surrogate duality for problem (P) above, by taking W = F* and EG,w = ty E F j w(Y) sup w(G)} (w E F*). 125) means that the distance from yo to G is equal to the maximum of the distances from yo to the closed half-spaces containing G and supporting G. This result of best approximation theory provides another motivation for studying surrogate duality. 8a. 127) where W is an arbitrary set and X = À G ' h : W —> R is any function.

14), so the closed subsets of X are "B-convex" subsets of X in the sense of the above definition. , see Hammer [110]—[113] for systematic applications of this idea). 15) form one of the classical systems of axioms defining a topological space). 15) is not satisfied by the convexity system B of all convex subsets of R". Another important class of examples of convexity systems is obtained in sets X endowed with some algebraic structure. 15)). 1a) is a convexity system. , B is a convexity system in X.

22) is used to define the "W-convexity" of Q by the condition that the W-convex hull of each compact subset of Q should be compact. , see Fuks [93]). , see Hiirmander [122], [123]). However, in the present book we will not consider complex convexity of sets and functions. The approach to abstract convexity of sets via hull operators is equivalent to the approach via convexity systems. 23) BEB GCB is a hull operator such that a set G c X is B-convex if and only if it is u-convex. Thus both approaches encompass the same particular cases.