Universal Polynomial Majorants on Convex Bodies Original Research Article

Let K be a convex body in Rd (d⩾2), and denote by Bn(K) the set of all polynomials pn in Rd of total degree ⩽n such that |pn|⩽1 on K. In this paper we consider the following question: does there exist a p*n∈Bn(K) which majorates every element of Bn(K) outside of K? In other words can we find a minimal γ⩾1 and p*n∈Bn(K) so that |pn(x)|⩽γ |p*n(x)| for every pn∈Bn(K) and x∈Rd\K? We discuss the magnitude of γ and construct the universal majorants p*n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.