Best Approximation by Normal and Conormal Sets

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 RI+ of a finite-dimensional coordinate space RI endowed with the max-norm. A normal (respectively, conormal) set arises as the set of all solutions of a system of inequalities fα(x)⩽0 (α∈A), x∈RI+ (respectively, fα(x)⩾0 (α∈A), x∈RI+), where fα is an increasing function and A is an arbitrary set of indices. We consider these sets as analogues (in 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.