Best approximation by normal and conormal sets

[摘要] The aim of the present paper is to develop a theory of best approximation by elements of so-called normal sets and their complements-conormal sets in the non-negative orthant R-+(I) of a finite-dimensional coordinate space R-I endowed with the max-norm. A normal respectively. conormal) set arises as the set of all solutions of a system of inequalities f(alpha)(x) less than or equal to 0 (alpha is an element ofA). x is an element of R-+(I) (respectively. f(alpha)(x) greater than or equal to 0 (alpha is an element of A). x is an element of R-+(I)), where f(alpha) is an increasing function and A is an arbitrary set of. indices. We consider there sets as analogues tin a certain sense) of convex sets, and we use the so-called min-type functions as analogues of linear functions. We show that many results on best approximation by convex and reverse convex sets and corresponding separation theory (but not all of them) have analogues in the case under consideration. At the same time there are no convex analogues for many results related to best approximation by normal sets. (C) 2000 Academic Press.

[发布日期] 2000-12-01