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POINT-SETS IN GENERAL POSITION WITH MANY SIMILAR COPIES OF A PATTERN
 

Summary: POINT-SETS IN GENERAL POSITION WITH MANY
SIMILAR COPIES OF A PATTERN
BERNARDO M. ŽABREGO AND SILVIA FERNŽANDEZ-MERCHANT
Department of Mathematics
California State University, Northridge
18111 Nordhoff St, Northridge, CA 91330-8313.
email:{bernardo.abrego,silvia.fernandez}@csun.edu
Abstract. For every pattern P, consisting of a finite set of
points in the plane, SP (n) is defined as the largest number of
similar copies of P among sets of n points in the plane without
3 points on a line. A general construction, based on iterated
Minkovski sums, is used to obtain new lower bounds for SP (n)
when P is an arbitrary pattern. Improved bounds are obtained
when P is a triangle or a regular polygon with few sides.
1. Introduction
Sets A and B in the plane are similar, denoted by A B, if
there is an orientation-preserving isometry followed by a dilation
that takes A to B. Identifying the plane with C, the set of complex
numbers, A B if there are complex numbers w and z = 0 such that
B = zA+w. Here, zA = {za : a A} and A+w = {a+w : a A}.

  

Source: Abrego, Bernardo - Department of Mathematics, California State University, Northridge
Fernandez, Silvia - Department of Mathematics, California State University, Northridge

 

Collections: Mathematics