An example of an "unlinked" set of 2k + 3 points in 2k-space

2024/02/14 by Starkov, M.
#52A20 #52A37 #52C35 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2402.09002

Abstract

Take any d + 3 points in ℝ^d. It is known that (a) if d = 2k + 1, then there are two linked (k + 1)-simplices with the vertices at these points; (b) if d = 2k, then there are two disjoint (k + 1)-tuples of these points such that their convex hulls intersect. The analogue of (b) for d = 2k + 1, which is also the analogue of (a) for intersections (instead of linkings), states that there are two disjoint (k + 1)- and (k + 2)-tuples of these points such that their convex hulls intersect. This analogue is correct by (a).

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