For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$ normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of $f( \mathbf{x})$ is given by \[ A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}-t\mathbf{u})d\mu(\mathbf{u}). \] It will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has the equivalence \[ \|A_tf-f\|_B\approx \inf \{ (\|f-g\|_B+t^2 \|P(D)g\|_B): P(D)g\in B\}\quad\text{for }t>0, \] where $\varphi(t)\approx \psi(t)$ means $c^{-1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions for which translations are continuous isometries and $P(D)$ is an elliptic differential operator induced by $\mu$. Many applications are given, notable among which is the averaging operator with $d\mu(\mathbf{u})= \frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of $S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate of approximation by averages on the boundary of a convex set under more restrictive conditions is also shown to be equivalent to an appropriate $K$-functional.

Bivariate polynomials with a fixed leading term $x^m y^n$, which deviate least from zero in the uniform or $L^2$-norm on the unit disk $D$ (resp. a triangle) are given explicitly. A similar problem in $L^p$, $1 \le p \le \infty$, is studied on $D$ in the set of products of linear polynomials.

Averages in weighted spaces $L^p_\phi[-1,1]$ defined by additions on $[-1,1]$ will be shown to satisfy strong converse inequalities of type A and B with appropriate $K$-functionals. Results for higher levels of smoothness are achieved by combinations of averages. This yields, in particular, strong converse inequalities of type D between $K$-functionals and suitable difference operators.