Publication: Geometry of Homogeneous Polynomials on non Symmetric Convex Bodies
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2009
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Matematisk Inst
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If Delta 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((2)Delta) of the 2-homogeneous polynomials on R(2) endowed with the norm given by parallel to ax(2) + bxy + cy(2)parallel to(Delta) := sup{vertical bar ax(2) + bxy + cy(2)vertical bar : (x, y) is an element of Delta} for every a, b, C E R. We investigate some geometrical properties of this norm. We provide an explicit formula for parallel to.parallel to(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 P((2)A) and show that a classical inequality of Martin does not remain true with the same constant for homogeneous polynomials on non symmetric convex bodies.
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