By A. S. Smogorzhevski

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The following simple observation can be useful to determine what a best approximating polynomial may look like. 1). Thus we have pi (x) ≤ x 2n ≤ C 2n pi (x) for all x ∈ V and for every pi . 1). Suppose now that the normed space V possesses a compact group G of linear isometries. 1). Hence if a norm · is invariant under the action of a compact group, we can always choose an invariant best approximating polynomial. 1. )−1/2n ≈ e/n. Since for any ﬁxed n, computation of p(x) takes a dO(n) time, for any c > 0 we obtain a polynomial time algorithm to approximate x within a factor 22 A.

Ark. D. (1983): A topological application of the isoperimetric inequality. Amer. J. , Simonovits, M. (1995): Isoperimetric problems for convex bodies and a localization lemma. Discrete and Comput. E. (1961): Moment inequalities of Polya frequency functions. Paciﬁc J. , 11, 1023–1033 Spectral Gap and Concentration for Probability Measures [L] [Le] [Pi] [Pr1] [Pr2] [S] 43 Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. , 1709, Springer, 120–216 Leindler, L. (1972): On a certain converse of H¨ older’s inequality II.

1). Thus we have pi (x) ≤ x 2n ≤ C 2n pi (x) for all x ∈ V and for every pi . 1). Suppose now that the normed space V possesses a compact group G of linear isometries. 1). Hence if a norm · is invariant under the action of a compact group, we can always choose an invariant best approximating polynomial. 1. )−1/2n ≈ e/n. Since for any ﬁxed n, computation of p(x) takes a dO(n) time, for any c > 0 we obtain a polynomial time algorithm to approximate x within a factor 22 A. Barvinok √ of c d (again, we do not count the time required for preprocessing, that is, to ﬁnd the polynomial p).