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Convex polyhedra in R3 spanning -(n4=3
 

Summary: Convex polyhedra in R3
spanning -(n4=3
) congruent triangles.
Bernardo M. ¶Abrego
California State University Northridge
bernardo.abrego@csun.edu
Silvia Fern¶andez-Merchant
California State University Northridge
silvia.fernandez@csun.edu
Abstract
We construct n-vertex convex polyhedra with the property stated in the title
In this note we construct, for every ¯xed triangle T, a n-vertex convex polyhedron
determining -(n4=3
) triangles congruent to T among its triplets. Even with the convexity
assumption dropped, this was only known when T is an isosceles right triangle (see [2] and
[4]). With respect to the upper bound Brass [2] proved that n points in R3 span at most
O(n7=4+"
) triangles congruent to T and very recently Agarwal and Sharir [1] improved this
and obtained the current best bound of O(n5=3+"). There are no better bounds that take
advantage of the convexity restriction.

  

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

 

Collections: Mathematics