Geodesic
Inequalities for Polynomials on the Unit Square via the Krein-Milman Theorem
Journal of convex analysis, Tome 20 (2013) no. 1, pp. 125-142 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 41A17, 26D05, 52A21, 46B04
Mots-clés : Convexity, extreme points, polynomial norms, Bernstein and Markov inequalities, polarization constants 
@article{JCA_2013_20_1_JCA_2013_20_1_a7,
     author = {J. L. G\'amez-Merino and G. A. Mu\~noz-Fern\'andez and V. M. S\'anchez and J. B. Seoane-Sep\'ulveda},
     title = {Inequalities for {Polynomials} on the {Unit} {Square} via the {Krein-Milman} {Theorem}},
     journal = {Journal of convex analysis},
     pages = {125--142},
     year = {2013},
     volume = {20},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2013_20_1_JCA_2013_20_1_a7/}
}
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J. L. Gámez-Merino; G. A. Muñoz-Fernández; V. M. Sánchez; J. B. Seoane-Sepúlveda. Inequalities for Polynomials on the Unit Square via the Krein-Milman Theorem. Journal of convex analysis, Tome 20 (2013) no. 1, pp. 125-142. http://geodesic.mathdoc.fr/item/JCA_2013_20_1_JCA_2013_20_1_a7/