Whitney estimates for convex domains with applications to multivariate piecewise polynomial approximation

We prove the following Whitney estimate. Given 0 < p less than or equal to infinity, r epsilon N, and d greater than or equal to 1, thereexists aconstant C(d, r, p), depending only on the three parameters, Such that for every bounded convex domain Omega subset of R-d, and each function f epsilon L-p (Omega), Er-1 (f, Omega), less than or equal to C(d r. P)omega(r)(f, diam(Omega))(p), where Er-1(f, Omega)(p) is the degree of approximation by polynomials of total degree, r - 1, and omega(r)(f, )(p) is the modulus of smoothness of order r. Estimates like this can be found in the literature but with constants that depend in an essential way on the geometry of the domain, in particular, the domain is assumed to be a Lipschitz domain and the above constant C depends on the minimal head-angle of the cones associated with the boundary. The estimates we obtain allow LIS to extend to the multivariate case, the results on bivariate Skinny B-spaces of Karaivanov and Petrushev on characterizing nonlinear approximation from nested triangulations. In a sense, our results were anticipated by Karaivanov and Petrushev.