Inequalities for Polynomials on the Unit Square via the Krein-Milman Theorem

We provide sharp Bernstein and Markov inequalities for 2-homogeneous polynomials on the square $\Box\subset {\mathbb R}^2$ with vertices $(0,0)$, $(1,0)$, $(1,1)$ and $(0,1)$. If ${\mathcal P}(^2\Box)$ is the space of such polynomials, we also find the polarization constant of ${\mathcal P}(^2\Box)$ and the unconditional constant for the canonical basis of ${\mathcal P}(^2\Box)$. All the results are obtained by means of the Krein-Milman Theorem, using a characterization of the extreme 2-homogeneous polynomials on $\Box$ which is also given in the paper.