Polynomial approximation on polytopes

Polynomial approximation on convex polytopes in d is considered in uniform and Lp-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the Lp -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.