Journal of Convex Analysis 20 (2013), No. 1, 125--142
Copyright 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.