Universal polynomial majorants on convex bodies (Q5945601)
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scientific article; zbMATH DE number 1657268
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Universal polynomial majorants on convex bodies |
scientific article; zbMATH DE number 1657268 |
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Universal polynomial majorants on convex bodies (English)
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10 July 2002
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polynomials
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convex body
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ellipsoid
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0.9091935
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0.88764113
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0.88660717
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0.8842028
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Denote by \(B_n(K)\) the set of all polynomials \(p_n\) of \(d\) variables and of degree at most \(n\) such that \(|p_n|\leq 1\) on a given convex body \(K\subset \mathbb{R}^d\).NEWLINENEWLINENEWLINEThe author proves that for every even \(n\) there exists \(p^*_n\in B_n(K)\) and a constant \(\gamma\geq 1\) such that for every \(p_n\in B_n(K)\) and for every \(x\in \mathbb{R}^d \setminus K\) we have \(|p_n(x) |\leq \gamma \cdot|p^*_n (x)|\). Moreover, if \(\gamma=1\), then \(K\) cannot be an ellipsoid.
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