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1. Chord measures in integral geometry and their Minkowski problems

   https://doi.org/10.1002/cpa.22190  

   Lutwak, Erwin; Xi, Dongmeng; Yang, Deane; Zhang, Gaoyong (July 2024, Communications on Pure and Applied Mathematics)

   Varadhan, S_R_Srinivasa (Ed.)

   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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2. The L _p chord Minkowski problem

   https://doi.org/10.1515/ans-2022-0041  

   Xi, Dongmeng; Yang, Deane; Zhang, Gaoyong; Zhao, Yiming (January 2023, 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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3. A geometric variational framework for computing spherical optimal transportation maps II

   https://doi.org/10.4310/MCGD.2022.v2.n2.a2  

   Zhao, Zhou; Lei, Na; Cui, Li; Su, Kehua; Xu, Xiaoyin; Luo, Feng; Gu, Xianfeng; Yau, Shing-Tung (January 2022, Mathematics, Computation and Geometry of Data)

   Optimal transportation maps play fundamental roles in many engineering and medical fields. The computation of optimal transportation maps can be reduced to solve highly non-linear Monge-Ampere equations. This work summarizes the geometric variational frameworks for spherical optimal transportation maps, which offers solutions to the Minkowski problem in convex differential geometry, reflector design and refractor design problems in optics. The method is rigorous, robust and efficient. The algorithm can directly generalized to higher dimensions. 

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4. Nesterov acceleration in benignly non-convex landscapes

   Gupta, Kanan; Wojtowytsch, Stephan (January 2025, ICLR)

   While momentum-based optimization algorithms are commonly used in the notoriously non-convex optimization problems of deep learning, their analysis has historically been restricted to the convex and strongly convex setting. In this article, we partially close this gap between theory and practice and demonstrate that virtually identical guarantees can be obtained in optimization problems with a ‘benign’ non-convexity. We show that these weaker geometric assumptions are well justified in overparametrized deep learning, at least locally. Variations of this result are obtained for a continuous time model of Nesterov’s accelerated gradient descent algorithm (NAG), the classical discrete time version of NAG, and versions of NAG with stochastic gradient estimates with purely additive noise and with noise that exhibits both additive and multiplicative scaling. 

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5. Extremal problems for convex geometric hypergraphs and ordered hypergraphs

   https://doi.org/10.4153/S0008414X20000632  

   Füredi, Zoltán; Jiang, Tao; Kostochka, Alexandr; Mubayi, Dhruv; Verstraëte, Jacques (December 2021, Canadian Journal of Mathematics)

   Abstract An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braß–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting. 

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