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The \(L_p\) chord Minkowski problem in a critical interval

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Abstract

Chord measures and \(L_p\) chord measures were recently introduced by Lutwak-Xi-Yang-Zhang by establishing a variational formula regarding a family of fundamental integral geometric invariants called chord integrals. Prescribing the \(L_p\) chord measure is known as the \(L_p\) chord Minkowski problem, which includes the \(L_p\) Minkowski problem heavily studied in the past 2 decades as special cases. In the current work, we solve the \(L_p\) chord Minkowski problem when \(0\le p<1\), without symmetry assumptions.

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Notes

  1. As a comparison, the classical Minkowski problem studies the surface area measure which is also known as the area measure \(S_{n-1}\).

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Acknowledgements

The authors are extremely grateful to the referees for their many valuable comments and suggestions. Research of Guo was supported, in part, by NSFC Grants 12126319 and 12126368. Research of Xi was supported, in part, by NSFC Grant 12071277 and STCSM Grant 20JC1412600. Research of Zhao was supported, in part, by NSF Grant DMS–2132330.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Henan, 453007, China

    Lujun Guo

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Dongmeng Xi

  3. Department of Mathematics, Syracuse University, Syracuse, 13244, NY, USA

    Yiming Zhao

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  1. Lujun Guo
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  2. Dongmeng Xi
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  3. Yiming Zhao
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Correspondence to Dongmeng Xi.

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Guo, L., Xi, D. & Zhao, Y. The \(L_p\) chord Minkowski problem in a critical interval. Math. Ann. 389, 3123–3162 (2024). https://doi.org/10.1007/s00208-023-02721-8

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