The Union of Colorful Simplices Spanned by a Colored Point Set

A simplex spanned by a colored point set in Euclidean d-space is colorful if all vertices have distinct colors. The union of all full-dimensional colorful simplices spanned by a colored point set is called the colorful union. We show that for every d N, the maximum combinatorial complexity of the colorful union of n colored points in Rd is between (n(d-1)2 ) and O(n(d-1)2 log n). For d = 2, the upper bound is known to be O(n), and for d = 3 we present an upper bound of O(n4 (n)), where (

• Type: Journal
• Year: 2010
• Where: COCOA
• Authors: André Schulz, Csaba D. Tóth
• Keywords: COCOA 2010, Colored Point, Colorful Union, Discrete Geometry, Full-dimensional Colorful Simplices