Are the Kantorovitch Polynomials Area Diminishing?

Abstract

The Bernstein-Bezier polynomials are known to possess total variation and length diminishing properties in one variable. We investigate the two dimensional generalizations to the square and the triangle. Simple counterexamples show that they do not diminish surface area. We consider Kantorovitch polynomials which seem to be a better choice to be area diminishing. A counterexample is given for the square. We then define the Kantorovitch polynomials on the triangle and give an area estimate for them.