It is our purpose in the present note to investigate the behavior of functions p(x) of best approximation to a given function f(x), where p(x) is supposed to belong to a linear family P and where deviation is measured on E:O O, w(x) = 1, and is now to be studied more generally (but not including the case p < 1). For positive r(t) we may write r (t) -= t(t), and our problem becomes that of variable-power approximation, an aspect that we do not stress for nonconstant a(t). We use r(t) directly to define our norm, in contrast to Orlicz who introduces a function comparable with our r(t) but defines quite differently a norm in the classical sense. We considered in reference' and consider particularly here properties of oscillation of f(x) - p(x) on E and identical vanishing of f(x) - p(x) on a subset of E of positive measure, extensions of classical results due to D. Jackson. Our previous results derived necessary conditions for best approximators, whereas here we treat also sufficient conditions. Under suitable conditions we prove the existence of a weight function so that a given p(x) shall be a best approximator. Our methods involve a certain algebraic inequality akin to orthogonality (inequality [1] below), of a type which was recently used by one of us2 in an invited address. The authors have previously investigated3 approximation on a finite set with the pth power norm, but postpone such consideration of the r-norm to another occasion. 1. We consider henceforth a deviation as defined by a generalized norm |g(x)|| =fEr(Ig(x) )w(x)dx, E:O < x < 1, where r(t) is defined for all t, 0 < t < o, where T(t) exists and is continuous for such t, and where w(x) is continuous on E and positive a.e. We study for a given function