Given a finite set A \subseteq R^d, points a_1,a_2,\dotsc,a_{\ell} \in A form an \ell-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  .
Given a finite set A \subseteq R^d, points a_1,a_2,\dotsc,a_{\ell} \in A form an \ell-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 O(4^dd\log d) or more. This improves the previously known upper bound of order d^{d+o(d)} due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences , yielding a bound of 2^{7d .
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2^{7d
O(4^dd\log d)
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Our
The basic version of our
A
, yielding a bound of 2^{7d