Sets in Rd with no large empty convex subsets

✍ Scribed by Pavel Valtr

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 648 KB
• Volume: 108
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)90665-3

✦ Synopsis

Let A be a finite set of points in general position in Rd, d 2 2. Points a,, a2, , a, E A form an n-hole in A (an empty convex subset of A) if they are vertices of a convex polytope containing no other point of A. Let h(d) denote the maximum number h such that any sufficiently large set of points in general position in IWd contains an h-hole. For any d 22, we show that 2d + 1 s h(d) s 2dm'(P(d -1) + l), where P(d -1) is the product of the smallest d -1 prime numbers. For d = 3 we show a better upper bound: h(3) c 22.