Equal area polygons in convex bodies

Authors:
T. Sakai;C. Nara;J. Urrutia
Affiliations:
Research Institute of Educational Development, Tokai University, Shibuya-ku, Tokyo, Japan;Research Institute of Educational Development, Tokai University, Shibuya-ku, Tokyo, Japan;Instituto de Matemáticas, Ciudad Universitaria, Universidad Nacional Autónoma de México, México D.F., México
Venue:
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
Year:
2003

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Abstract

In this paper, we consider the problem of packing two or more equal area polygons with disjoint interiors into a convex body K in E2 such that each of them has at most a given number of sides. We show that for a convex quadrilateral K of area 1, there exist n internally disjoint triangles of equal area such that the sum of their areas is at least $\frac {4n} {4n+1}$. We also prove results for other types of convex polygons K. Furthermore we show that in any centrally symmetric convex body K of area 1, we can place two internally disjoint n-gons of equal area such that the sum of their areas is at least $\frac {n-1}{\pi} sin \frac {\pi}{n-1}$. We conjecture that this result is true for any convex bodies.