Approximating a Norm by a Polynomial

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.