Journal of Convex Analysis 21 (2014), No. 3, 745--764
Copyright Heldermann Verlag 2014
Supremum Norms for 2-Homogeneous Polynomials on Circle Sectors
Gustavo A. Muñoz-Fernández
Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain
gustavo_fernandez@mat.ucm.es
Daniel Pellegrino
Dep. de Matemática,, Universidade Federal da Paraíba, 58.051-900 - João Pessoa, Brazil
pellegrino@pq.cnpq.br
Juan B. Seoane-Sepúlveda
Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain
jseoane@mat.ucm.es
Andreas Weber
Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstr. 89-93, 76128 Karlsruhe, Germany
andreasweber.mail@gmail.com
[Abstract-pdf]
We consider the Banach space of two homogeneous polynomials endowed with the supremum norm $\|\cdot\|_{D(\beta)}$ over circle sectors $D(\beta)$ of angle $\beta$ for several values of $\beta\in[0,2\pi]$. We provide an explicit formula for $\|\cdot\|_{D(\beta)}$, a full description of the extreme points of the corresponding unit balls, and a parametrization and a plot of their unit spheres. This work is an extension of a series of papers on the same topic published in the last decade and it has a number of applications to obtain polynomial-type inequalities.
Keywords: Bernstein and Markov inequalities, unconditional constants, polarizations constants, polynomial inequalities, homogeneous polynomials, extreme points.
MSC: 46G25; 46B28, 41A44
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