Uniform approximation by quasi-convex and convex functions

Abstract

Given a bounded function f defined on a convex subset of R n , the two problems considered are to find a quasi-convex (convex) function which is a best approximation to f under the uniform norm. It is shown that if f is the greatest quasi-convex (convex) minorant of f, then f′ = f + c , for some c ≧ 0, is the maximal best quasi-convex (convex) approximation to f. Furthermore, the nonlinear operator T defined by T( f) = f′ is a Lipschitzian selection operator with some least constant C( T), where C( T) ≤ C( T′) for all Lipschitzian operators T′ which map f to one of its best approximations. Thus T is optimal in this sense.