On convex holes in d-dimensional point sets

07/17/2020

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by   Boris Bukh, et al.

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Given a finite set A ⊆ℝ^d, points a_1,a_2,…,a_ℓ∈ A form an ℓ-hole in A if they are the vertices of a convex polytope which contains no points of A in its interior. We construct arbitrarily large point sets in general position in ℝ^d having no holes of size 2^7d or more. This improves the previously known upper bound of order d^d+o(d) due to Valtr. Our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences.