Approximating a norm by a polynomial

الأصل · EN

We prove that for any norm |*| 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 p¹/²ⁿ(x) < |x| < c(n,d) p¹/²ⁿ(x), where c(n,d)=n+d-1 choose n¹/²ⁿ. Corollaries and polynomial approximations of the Minkowski functional of a convex body are discussed.