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Intersection theoretic inequalities via Lorentzian polynomials

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Abstract

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to m-positive classes and Schur classes. We also study its convexity variants—the geometric inequalities for m-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by m-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov–Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Plünnecke–Ruzsa inequalities) from additive combinatorics in our contexts.

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Acknowledgements

This work is supported by the National Key Research and Development Program of China (No. 2021YFA1002300) and National Natural Science Foundation of China (No. 11901336). We would like to thank Julius Ross for kindly sharing the work on dually Lorentzian polynomials with us. We also thank the referee for the careful reading and helpful comments.

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  1. Tsinghua University, Beijing, 100084, China

    Jiajun Hu & Jian Xiao

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  1. Jiajun Hu
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Correspondence to Jian Xiao.

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In memory of Jean-Pierre Demailly.

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Hu, J., Xiao, J. Intersection theoretic inequalities via Lorentzian polynomials. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02822-y

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