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Archiv der Mathematik

, Volume 113, Issue 1, pp 63–72 | Cite as

Plurisubharmonic geodesics and interpolating sets

  • Dario Cordero-Erausquin
  • Alexander RashkovskiiEmail author
Article
  • 21 Downloads

Abstract

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of \({\mathbb {C}}^n\). Namely, two non-pluripolar, polynomially closed, compact subsets of \({\mathbb {C}}^n\) are interpolated as level sets \(L_t=\{z: u_t(z)=-1\}\) for the geodesic \(u_t\) between their relative extremal functions with respect to any ambient bounded domain. The sets \(L_t\) are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge–Ampère capacities of \(L_t\) satisfy a dual Brunn–Minkowski inequality.

Keywords

Complex interpolation Plurisubharmonic geodesic Relative extremal function Monge–Ampère capacity Brunn–Minkowski inequality 

Mathematics Subject Classification

Primary 32E30 Secondary 32U05 32U15 32U20 52A20 52A38 52A40 

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Notes

Acknowledgements

Part of the work was done while the second named author was visiting Université Pierre et Marie Curie in March 2017; he thanks Institut de Mathématiques de Jussieu for the hospitality. The authors are grateful to the anonymous referee for careful reading of the text.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dario Cordero-Erausquin
    • 1
  • Alexander Rashkovskii
    • 2
    Email author
  1. 1.Institut de Mathématiques de JussieuSorbonne UniversitéParis Cedex 05France
  2. 2.Tek/NatUniversity of StavangerStavangerNorway

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