Polynomial Approximation on Polytopes

Polynomial approximation on convex polytopes in $\mathbf{R}^d$ is considered in uniform and $L^p$-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the $L^p$-case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate $K$-functional follows as a consequence. The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.