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On the \(L^{p}\) dual Minkowski problem for \(-1<p<0\)

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Abstract

The \(L^{p}\) dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the \(L^{p}\) dual Minkowski problem, asks about existence of a convex body with prescribed \(L^{p}\) dual curvature measure. This question unifies the previously disjoint \(L^{p}\) Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the \(L^{p}\) dual Minkowski problem for the case of \(q<p+1\), \(-1<p<0\), and \(p\ne q\) for even measures.

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. DMS-2005875.

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  1. Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Stephanie Mui

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  1. Stephanie Mui
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Mui, S. On the \(L^{p}\) dual Minkowski problem for \(-1<p<0\). Calc. Var. 63, 215 (2024). https://doi.org/10.1007/s00526-024-02806-5

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