Geometry of homogeneous polynomials on non symmetric convex bodies

Authors

  • G. A. Muñoz-Fernández
  • S. Gy. Révész
  • J. B. Seoane-Sepúlveda

DOI:

https://doi.org/10.7146/math.scand.a-15111

Abstract

If $\Delta$ stands for the region enclosed by the triangle in ${\mathsf R}^2$ of vertices $(0,0)$, $(0,1)$ and $(1,0)$ (or simplex for short), we consider the space ${\mathcal P}(^2\Delta)$ of the 2-homogeneous polynomials on ${\mathsf R}^2$ endowed with the norm given by $\|ax^2+bxy+cy^2\|_\Delta:=\sup\{|ax^2+bxy+cy^2|:(x,y)\in\Delta\}$ for every $a,b,c\in{\mathsf R}$. We investigate some geometrical properties of this norm. We provide an explicit formula for $\|\cdot\|_\Delta$, a full description of the extreme points of the corresponding unit ball and a parametrization and a plot of its unit sphere. Using this geometrical information we also find sharp Bernstein and Markov inequalities for ${\mathcal P}(^2\Delta)$ and show that a classical inequality of Martin does not remain true for homogeneous polynomials on non symmetric convex bodies.

Downloads

Published

2009-09-01

How to Cite

Muñoz-Fernández, G. A., Révész, S. G., & Seoane-Sepúlveda, J. B. (2009). Geometry of homogeneous polynomials on non symmetric convex bodies. MATHEMATICA SCANDINAVICA, 105(1), 147–160. https://doi.org/10.7146/math.scand.a-15111

Issue

Section

Articles