Approximating a norm by a polynomial
Abstract
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.
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