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Universal Polynomial Majorants on Convex Bodies

✍ Scribed by András Kroó

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
148 KB
Volume
111
Category
Article
ISSN
0021-9045

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✦ Synopsis

Let K be a convex body in R d (d 2), and denote by B n (K) the set of all polynomials p n in R d of total degree n such that | p n | 1 on K. In this paper we consider the following question: does there exist a p n * # B n (K) which majorates every element of B n (K) outside of K? In other words can we find a minimal # 1 and p n * # B n (K) so that | p n (x)| # | p n *(x)| for every p n # B n (K) and x # R d "K? We discuss the magnitude of # and construct the universal majorants p n * for even n. It is shown that # can be 1 only on ellipsoids. Moreover, #=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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✍ András Kroó 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 106 KB

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