Stone-Weierstrass theorem for homogeneous polynomials and its role in convex geometry

2020/12/09 by Merino, Bernardo González, Villa, Rafael
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2012.04999

Abstract

We give a uniform approximation of the characteristic function of the boundary of a centrally symmetric n-dimensional compact and convex set by homogeneous polynomials of even degree d fulfilling |g_d-1|≤ E/d^1/2-β, for every β>0, large enough d, and some constant E only depending on n and K. In particular, this proves a conjecture posed by Kroo in 2004, also known as the Stone-Weierstrass theorem for homogeneous polynomials. Moreover, we introduce the d-volume ratio for a convex body K in \mathbb Rⁿ, by means of its d-Lasserre-Löwner polynomial. We also prove an upper bound of the d-volume ratio of the form 1+F/d^3/2-β, for every β>0, large enough d, and F some constant only depending on n.

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