Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 20 (2013), No. 1, 125--142Copyright Heldermann Verlag 2013 Inequalities for Polynomials on the Unit Square via the Krein-Milman Theorem José Luis Gámez-Merino Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain jlgamez@mat.ucm.es Gustavo A. Muñoz-Fernández Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain gustavo_fernandez@mat.ucm.es Viktor M. Sánchez Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain victorms@mat.ucm.es Juan B. Seoane-Sepúlveda Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain jseoane@mat.ucm.es [Abstract-pdf] 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. Keywords: Convexity, extreme points, polynomial norms, Bernstein and Markov inequalities, polarization constants. MSC: 41A17; 26D05, 52A21, 46B04 [ Fulltext-pdf  (668  KB)] for subscribers only.