Polynomial Approximation on Polytopes

Polynomial approximation on convex polytopes in     is considered in uniform and    -norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the    -case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate  -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.