Approximating a Norm by a Polynomial

Abstract

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.

This research was partially supported by NSF Grant DMS 9734138.