Radon Numbers Grow Linearly

Define the k -th Radon number r k of a convexity space as the smallest number (if it exists) for which any set of r k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove...

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Bibliographic Details
Published inDiscrete & computational geometry Vol. 68; no. 1; pp. 165 - 171
Main Author Pálvölgyi, Dömötör
Format Journal Article
LanguageEnglish
Published New York Springer US 01.07.2022
Springer Nature B.V
Subjects
Combinatorics
Computational Mathematics and Numerical Analysis
Convexity
Geometry
Hulls
Mathematics
Mathematics and Statistics
Radon
Convexity space
Radon numbers
Helly theorem
Tverberg theorem
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ISSN0179-5376
1432-0444
1432-0444
DOI10.1007/s00454-021-00331-2

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Summary:Define the k -th Radon number r k of a convexity space as the smallest number (if it exists) for which any set of r k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r k grows linearly, i.e., r k ≤ c ( r 2 ) · k .
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ISSN:0179-5376
1432-0444
1432-0444
DOI:10.1007/s00454-021-00331-2