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Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies

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Abstract

We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in \({\mathbb {R}}^d\), and by a tangential Markov inequality via an estimate of the Dubiner distance on smooth convex bodies. These allow to compute a \((1-\varepsilon )\)-approximation to the minimum of any polynomial of degree not exceeding n by \({\mathcal {O}}\left( (n/\sqrt{\varepsilon })^{\alpha d}\right) \) samples, with \(\alpha =2\) in the general case, and \(\alpha =1\) in the smooth case. Such constructions are based on three cornerstones of convex geometry, Bieberbach volume inequality and Leichtweiss inequality on the affine breadth eccentricity, and the Rolling Ball Theorem, respectively.

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Acknowledgements

Work partially supported by the DOR funds and the biennial project BIRD163015 of the University of Padova, and by the GNCS-INdAM. This research has been accomplished within the RITA “Research ITalian network on Approximation”

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  1. Department of Mathematics, University of Padova, Padova, Italy

    Federico Piazzon & Marco Vianello

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  1. Federico Piazzon
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  2. Marco Vianello
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Correspondence to Marco Vianello.

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Piazzon, F., Vianello, M. Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies. Optim Lett 13, 1325–1343 (2019). https://doi.org/10.1007/s11590-018-1377-0

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  • DOI: https://doi.org/10.1007/s11590-018-1377-0

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