Geometry of homogeneous polynomials on non symmetric convex bodies

Autores: Gustavo Adolfo Muñoz Fernández , Szilárd Gy Révész, Juan Benigno Seoane Sepúlveda
Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 105, Nº 1, 2009, págs. 147-160
Idioma: inglés
DOI: 10.7146/math.scand.a-15111
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Resumen
- 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.