BEST MONOTONE L_■-APPROXIMATIONS IN SEVERAL VARIABLES

Given a continuous function f defined on the unit cube of R~n and a convexfunction _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set ofbest L~(t)-approximations by monotone functions has exactly one elementft,which is also a continuous function.Moreover if the family of convexfunctions {_t}t>0 converges uniformly on compact sets to a function _0,then the best approximation f_t→f_0 uniformly,as t→0,where fo is thebest approximation of f within the Orlicz space L~(0) The best approxima-tions{f_t}are obtained as well as minimizing integrals or the Luxemburgnorm