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The Minkowski problem in Heisenberg groups

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Bin Chen, Juan Zhang, Peibiao Zhao, Xia Zhao

Differential Geometry · Submitted 3 Sep 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In the present paper, we develop for the first time a sub-Riemannian version of Minkowski type problem in the horizontal distributions of Heisenberg groups, and further give a positive answer to this sub-Riemannian Minkowski type problem via the variational method.

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arXiv ID2309.01170 (version 1)
Pith ID72XDEYV2, Pith's permanent record of this paper and its review
CitationBin Chen, Juan Zhang, Peibiao Zhao, Xia Zhao. "The Minkowski problem in Heisenberg groups." arXiv:2309.01170 (2023). https://pith.science/paper/72XDEYV2
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,694 input tokens.

What the abstract itself states, as probabilities: claims first 0.96 · states limitation 0.09 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.43 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

As we all know, the Minkowski type problem is the cornerstone of the Brunn-Minkowski theory in Euclidean space. The Heisenberg group as a sub-Riemannian space is the simplest non-Abelian degenerate Riemannian space that is completely different from a Euclidean space. By analogy with the Minkowski type problem in Euclidean space, the Minkowski type problem in Heisenberg groups is still open. In the present paper, we develop for the first time a sub-Riemannian version of Minkowski type problem in the horizontal distributions of Heisenberg groups, and further give a positive answer to this sub-Riemannian Minkowski type problem via the variational method.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 As we all know, the Minkowski type problem is the cornerstone of the Brunn-Minkowski theory in Euclidean space. background 1.00 0.03 0.01 asserted 1.00
s1 The Heisenberg group as a sub-Riemannian space is the simplest non-Abelian degenerate Riemannian space that is completely different from a Euclidean space. background 1.00 0.04 0.01 asserted 1.00
s2 By analogy with the Minkowski type problem in Euclidean space, the Minkowski type problem in Heisenberg groups is still open. gap 0.98 0.05 0.04 asserted 1.00
s3 In the present paper, we develop for the first time a sub-Riemannian version of Minkowski type problem in the horizontal distributions of Heisenberg groups, and further give a positive answer to this sub-Riemannian Minkowski type problem via the variational method. contribution 0.99 0.97 0.96 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

As we all know, the Minkowski type problem is the cornerstone of the Brunn-Minkowski theory in Euclidean space. The Heisenberg group as a sub-Riemannian space is the simplest non-Abelian degenerate Riemannian space that is completely different from a Euclidean space. By analogy with the Minkowski type problem in Euclidean space, the Minkowski type problem in Heisenberg groups is still open. In the present paper, we develop for the first time a sub-Riemannian version of Minkowski type problem in the horizontal distributions of Heisenberg groups, and further give a positive answer to this sub-Riemannian Minkowski type problem via the variational method.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites

Cites as background 1 paper
  • With the development of Brunn-Minkowski theory in Euclidean setting, Li and Xu in a recent seminal article [ref] started the study of Brunn-Minkowski theory in hyperbolic space, and solved the Horospherical Minkowski problem and the more general Horospherical Christoffel-Minkowski problem based on the concept of hyperbolic $p$-sum.

    Hyperbolic $p$-sum and Horospherical $p$-Brunn-Minkowski theory in hyperbolic space · An overview of the Minkowski problem · 0.89

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