Shape-Preserving Approximation by Real Multivariate Polynomials

Extending the results in the univariate case, in this chapter we prove approximation results preserving multivariate shapes, by Bernstein-type, convolutiontype, and tensor-product-type polynomials. The multivariate shapes considered are Popoviciu convexity, Schur convexity, axial convexity, polyhedral convexity, usual multivariate convexity, subharmonicity, and in general, L-convexity, with L a bounded linear operator satisfying some suitable conditions.