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Creators/Authors contains: "Xi, Dongmeng"

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  1. ; ; ; (, Journal of Differential Geometry)
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  2. ; ; (, Mathematische Annalen)
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  3. ; ; ; (, 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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  4. ; ; ; (, Communications on Pure and Applied Mathematics)
    Abstract To the families of geometric measures of convex bodies (the area measures of Aleksandrov‐Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions. 
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  5. ; (, International Mathematics Research Notices)
    Abstract General affine invariances related to Mahler volume are introduced. We establish their affine isoperimetric inequalities by using a symmetrization scheme that involves a total of $2n$ elaborately chosen Steiner symmetrizations at a time. The necessity of this scheme, as opposed to the usual Steiner symmetrization, is demonstrated with an example (see the Appendix). 
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  6. ; ; (, Advances in Mathematics)
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