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

  • Federico Piazzon
  • Marco Vianello
Original Paper
  • 6 Downloads

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.

Keywords

Polynomial optimization Norming mesh Markov inequality Tangential Markov inequality Dubiner distance Convex bodies 

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PadovaPadovaItaly

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