Resumen de Geometry of homogeneous polynomials on non symmetric convex bodies

Gustavo Adolfo Muñoz Fernández , Szilárd Gy Révész, Juan Benigno Seoane Sepúlveda

• If Δ stands for the region enclosed by the triangle in R2 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 R2 endowed with the norm given by ∥ax2+bxy+cy2∥Δ:=sup{|ax2+bxy+cy2|:(x,y)∈Δ} for every a,b,c∈R. 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.