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Geometry of spaces of homogeneous trinomials on \({\mathbb {R}}^2\)

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Abstract

For each pair of numbers \(m,n\in {{\mathbb {N}}}\) with \(m>n\), we consider the norm on \({{\mathbb {R}}}^3\) given by \(\Vert (a,b,c)\Vert _{m,n}=\sup \{|ax^m+bx^{m-n}y^n+cy^m|:x,y\in [-1,1]\}\) for every \((a,b,c)\in {{\mathbb {R}}}^3\). We investigate some geometrical properties of these norms. We provide an explicit formula for \(\Vert \cdot \Vert _{m,n}\), a full description of the extreme points of the corresponding unit balls and a parametrization and a plot of their unit spheres for certain values of m and n.

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Acknowledgements

G. A. Muñoz-Fernández and D. L. Rodríguez-Vidanes were supported by PGC2018-097286-B-I00. D. L. Rodríguez-Vidanes was also supported by the Spanish Ministry of Science, Innovation and Universities and the European Social Fund through a “Contrato Predoctoral para la Formación de Doctores, 2019” (PRE2019-089135).

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Authors and Affiliations

  1. Universidad de Valladolid, Campus Duques de Soria, 42004, Soria, Spain

    Pablo Jiménez-Rodríguez

  2. Facultad de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar y Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040, Madrid, Spain

    Gustavo A. Muñoz-Fernández & Daniel L. Rodríguez-Vidanes

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  1. Pablo Jiménez-Rodríguez
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  2. Gustavo A. Muñoz-Fernández
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Correspondence to Gustavo A. Muñoz-Fernández.

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Communicated by Ti-Jun Xiao.

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Jiménez-Rodríguez, P., Muñoz-Fernández, G.A. & Rodríguez-Vidanes, D.L. Geometry of spaces of homogeneous trinomials on \({\mathbb {R}}^2\). Banach J. Math. Anal. 15, 61 (2021). https://doi.org/10.1007/s43037-021-00144-8

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