Approximating a Norm by a Polynomial

  • Alexander BarvinokEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 1807)


We prove that for any norm \(\Vert \cdot \Vert\) in the d-dimensional real vector space V and for any odd n > 0 there is a non-negative polynomial p(x), \(x \in V\) of degree 2n such that
$$p^{1\over 2n}(x) \leq \Vert x\Vert \leq {n + d-1 \choose n}^{1 \over 2n} p^{1\over 2n}(x).$$
Corollaries and polynomial approximations of the Minkowski functional of a convex body are discussed.

Mathematics Subject Classification (2000):

46-06 46B07 52-06 60-06 


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Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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