Geometry of homogeneous polynomials on non symmetric convex bodies - Muñoz-Fernández, G. A.; Révész, S. Gy.; Seoane-Sepúlveda, J. B. (MATHEMATICA SCANDINAVICA, 2009)
If Δ stands for the region enclosed by the triangle in R^2 of vertices (0,0), (0,1) and (1,0) (or simplex for short), we consider the space P(^2Δ) of the 2-homogeneous polynomials on R^2 endowed with the norm given by \|ax^2+bxy+cy^2\|_Δ:={|ax^2+bxy+cy^2|:(x,y)Δ} for every a,b,cR. We investigate some geometrical properties of this norm. We provide an explicit formula for \|·\|_Δ, 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 P(^2Δ) and show that a classical inequality of Martin does not remain true for homogeneous polynomials on non symmetric convex bodies.