Universal Polynomial Majorants on Convex Bodies |
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Authors: | Andrs Kro |
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Institution: | Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13–15, H-1053, Hungary |
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Abstract: | Let K be a convex body in
d (d2), and denote by Bn(K) the set of all polynomials pn in
d of total degree n such that |pn|1 on K. In this paper we consider the following question: does there exist a p*nBn(K) which majorates every element of Bn(K) outside of K? In other words can we find a minimal γ1 and p*nBn(K) so that |pn(x)|γ |p*n(x)| for every pnBn(K) and x
d\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. |
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Keywords: | convex bodies polynomial majorants polytopes polytopal approximation |
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