A generalization of Bang's lemma

Gergely Ambrus

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A generalization of Bang's lemma

Gergely Ambrus

Metric Geometry · Submitted 21 Jan 2022

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

The authors' main claim We prove a common extension of Bang's and Kadets' lemmas for contact pairs, in the spirit of the Colourful Carath\'eodory Theorem.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,486 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.70 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.62 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We prove a common extension of Bang's and Kadets' lemmas for contact pairs, in the spirit of the Colourful Carath\'eodory Theorem. We also formulate a generalized version of the affine plank problem and prove it under special assumptions. In particular, we obtain a generalization of Kadets' theorem. Finally, we give applications to problems regarding translative and homothetic coverings.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove a common extension of Bang's and Kadets' lemmas for contact pairs, in the spirit of the Colourful Carath\'eodory Theorem.	contribution 0.84	0.96	0.70	asserted 1.00
s1	We also formulate a generalized version of the affine plank problem and prove it under special assumptions.	contribution 0.97	0.96	0.14	asserted 1.00
s2	In particular, we obtain a generalization of Kadets' theorem.	result 0.90	0.95	0.22	asserted 1.00
s3	Finally, we give applications to problems regarding translative and homothetic coverings.	contribution 0.92	0.95	0.07	asserted 0.99

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

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arXiv listing abstract v4

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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

We prove a common extension of Bang's and Kadets' lemmas for contact pairs, in the spirit of the Colourful Carath\'eodory Theorem. We also formulate a generalized version of the affine plank problem and prove it under special assumptions. In particular, we obtain a generalization of Kadets' theorem. Finally, we give applications to problems regarding translative and homothetic coverings.

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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

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