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Valuations in affine convex geometry

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Jakob Henkel, Thomas Wannerer

Metric Geometry · Submitted 26 Apr 2023

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

The authors' main claim We discover a surprising dichotomy: There are no new examples if one assumes translation invariance, but a plethora of examples without this assumption.

how it is stated · asserted 1.00 read the record →

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arXiv ID2304.13600 (version 3)
Pith IDQO2YU2CQ, Pith's permanent record of this paper and its review
CitationJakob Henkel, Thomas Wannerer. "Valuations in affine convex geometry." arXiv:2304.13600 (2023). https://pith.science/paper/QO2YU2CQ
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,986 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.66 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.61 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous; (3) finitely additive, and the resulting convex bodies are subsets of an irreducible representation of the special linear group. In this paper we explore the question whether there exist other constructions with these properties. We discover a surprising dichotomy: There are no new examples if one assumes translation invariance, but a plethora of examples without this assumption.

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 In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous; (3) finitely additive, and the resulting convex bodies are subsets of an irreducible representation of the special linear group. background 1.00 0.08 0.04 asserted 1.00
s1 In this paper we explore the question whether there exist other constructions with these properties. contribution 0.56 0.33 0.03 unclear 0.68
s2 We discover a surprising dichotomy: There are no new examples if one assumes translation invariance, but a plethora of examples without this assumption. result 1.00 0.96 0.94 asserted 1.00

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

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous; (3) finitely additive, and the resulting convex bodies are subsets of an irreducible representation of the special linear group. In this paper we explore the question whether there exist other constructions with these properties. We discover a surprising dichotomy: There are no new examples if one assumes translation invariance, but a plethora of examples without this assumption.

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2 arXiv papers cite this in 2 sentences

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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