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  1. ; (, Transactions of the American Mathematical Society)
    The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Ampère equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and sufficient condition for existence and uniqueness) in dimension 2 is presented. In higher dimensions, partial results are demonstrated. 
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    Free, publicly-accessible full text available May 29, 2026
  2. ; ; ; (, Calculus of Variations and Partial Differential Equations)
    Free, publicly-accessible full text available March 1, 2026
  3. ; ; ; (, Journal of Differential Geometry)
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  4. ; ; ; (, Advances in Mathematics)
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  5. ; ; (, Mathematische Annalen)
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  6. ; ; ; (, Advanced Nonlinear Studies)
    Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 p\gt 1 and the symmetric case of 0 < p < 1 0\lt p\lt 1 . 
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