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The (Self-Similar, Variational) Rolling Stones

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Dylan Langharst, Jacopo Ulivelli

Functional Analysis · Submitted 5 Apr 2023

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

The authors' main claim In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity.

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arXiv ID2304.02548 (version 5)
Pith IDCYDJZFJC, Pith's permanent record of this paper and its review
CitationDylan Langharst, Jacopo Ulivelli. "The (Self-Similar, Variational) Rolling Stones." arXiv:2304.02548 (2023). https://pith.science/paper/CYDJZFJC
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,688 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.70 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

The interplay between variational functionals and the Brunn-Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from B\"or\"oczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems \`a la Firey's worn stone problem.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The interplay between variational functionals and the Brunn-Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. background 1.00 0.03 0.02 asserted 1.00
s1 In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. result 0.73 0.96 0.73 asserted 1.00
s2 In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from B\"or\"oczky, Lutwak, Yang, and Zhang. contribution 0.55 0.86 0.17 asserted 0.99
s3 We show how these results imply the existence of self-similar solutions to variational flow problems \`a la Firey's worn stone problem. interpretation 0.78 0.92 0.26 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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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

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Original abstract

The interplay between variational functionals and the Brunn-Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from B\"or\"oczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems \`a la Firey's worn stone problem.

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