Bodies of Constant Width

Martini, Horst; Montejano, Luis; Oliveros, Déborah

doi:10.1007/978-3-030-03868-7

Cited by 71 publications

(24 citation statements)

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“…Concerning bodies of constant width in general, we refer to the recent comprehensive monograph [5] by Martini, Montejano, and Oliveros. Information on bodies of constant brightness can be found in Gardner's book [3], in particular Section 3.2 and its notes.…”

Section: Introductionmentioning

confidence: 99%

Integral geometry of pairs of hyperplanes or lines

Hug

Schneider

2020

Arch. Math.

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Crofton's formula of integral geometry evaluates the total motion invariant measure of the set of k-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and of constant brightness in the second case.

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Section: Introductionmentioning

confidence: 99%

Integral geometry of pairs of hyperplanes or lines

Hug

Schneider

2020

Arch. Math.

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“…Many well-known mathematicians such as W. Blaschke, M. Fujiwara, H. Lebesgue, K. Reidemeister, L.A. Santaló and W. Süss have made contribution to this field. Nevertheless, up to now some basic problems about convex bodies of constant width are still open (see [7], [31]). The next result is useful for Borsuk's problem.…”

Section: Reductions and Positive Resultsmentioning

confidence: 99%

“…In any metric space, Borsuk's corresponding problem can be reformulated in a similar way. Basic results about sets of constant width in general metric spaces can be found in Martini, Montejano and Oliveros[31]. Let β be a given positive number, and let K 1…”

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confidence: 99%

Borsuk’s partition conjecture

Zong

2021

Jpn. J. Math.

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In 1933, Borsuk proposed the following problem: Can every bounded set in E n be divided into n+1 subsets of smaller diameter? This problem has been studied by many authors, and a lot of partial results have been discovered. In particular, Kahn and Kalai's counterexamples surprised the mathematical community in 1993. Nevertheless, the problem is still far away from being completely resolved. This paper presents a broad review on related subjects and, based on a novel reformulation, introduces a computer proof program to deal with this challenging problem.

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“…Many well-known mathematicians such as W. Blaschke, M. Fujiwara, H. Lebesgue, K. Reidemeister, L. A. Santaló and W. Süss have made contribution to this field. Nevertheless, up to now some basic problems about convex bodies of constant width are still open (see [6,29]). The next result is useful for Borsuk's problem.…”

Section: Reductions and Positive Resultsmentioning

confidence: 99%

A Computer Program for Borsuk's Conjecture

Zong

2020

Preprint

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In 1933, Borsuk proposed the following problem: Can every bounded set in E n be divided into n+1 subsets of smaller diameters? This problem has been studied by many authors, and a lot of partial results have been discovered. In particular, Kahn and Kalai's counterexamples surprised the mathematical community in 1993. Nevertheless, the problem is still far away from being completely resolved. This paper presents a broad review on related subjects and, based on a novel reformulation, introduces a computer proof program to deal with this well-known problem.

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Bodies of Constant Width

Cited by 71 publications

References 0 publications

Integral geometry of pairs of hyperplanes or lines

Integral geometry of pairs of hyperplanes or lines

Borsuk’s partition conjecture

A Computer Program for Borsuk's Conjecture

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