Geometry of homogeneous polynomials on non symmetric convex bodies

G. A. Muñoz-Fernández, S. Gy. Révész, J. B. Seoane-Sepúlveda
2009 Mathematica Scandinavica  
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
more » ... ion 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.
doi:10.7146/math.scand.a-15111 fatcat:fntfp43hhber7bvm5l4bu5whxq