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.
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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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Cordero-Erausquin, D., Rashkovskii, A. Plurisubharmonic geodesics and interpolating sets. Arch. Math. 113, 63–72 (2019). https://doi.org/10.1007/s00013-018-01297-z
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DOI: https://doi.org/10.1007/s00013-018-01297-z
Keywords
- Complex interpolation
- Plurisubharmonic geodesic
- Relative extremal function
- Monge–Ampère capacity
- Brunn–Minkowski inequality