On Borsuk’s Conjecture for Two-Distance Sets

In this paper, we answer Larman’s question on Borsuk’s conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk’s conjecture is know...

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Bibliographic Details
Published in:Discrete & computational geometry 2014-04, Vol.51 (3), p.509-515
Main Author: Bondarenko, Andriy
Format: Article
Language:English
Subjects:
Combinatorics
Computational geometry
Computational Mathematics and Numerical Analysis
Geometry
Mathematical problems
Mathematics
Mathematics and Statistics
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Summary:In this paper, we answer Larman’s question on Borsuk’s conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk’s conjecture is known to be false. Other examples of two-distance sets with large Borsuk numbers are given.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-014-9579-4