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Empty Simplices in Euclidean Space

Published online by Cambridge University Press:  20 November 2018

Imre Bárány  and
Zoltán Füredi
Imre Bárány
Affiliation:
Or, Cornell University Ithaca, NY 14853
Zoltán Füredi
Affiliation:
Rutcor, Rutgers University New Brunswick, NJ 08903
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Abstract

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Let P - {p1,p2,. . . ,pn} be an independent point-set in ℝd (i.e., there are no d + 1 on a hyperplane). A simplex determined by d + 1 different points of P is called empty if it contains no point of P in its interior. Denote the number of empty simplices in P by fd(P). Katchalski and Meir pointed out that . Here a random construction Pn is given with , where K(d) is a constant depending only on d. Several related questions are investigated.

Keywords

Primary 52A37Secondary 10K30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 4 , 01 December 1987 , pp. 436 - 445
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Bárány, I., A generalization of Carathéodory's theorem, Discrete Math. 40 (1982), pp. 141 — 152.Google Scholar
2. Boros, E. and Füredi, Z., The number of triangles covering the center of an n-set, Geometriae Dedicata 17(1984), pp. 69– 77.Google Scholar
3. Danzer, L., Grünbaum, B. and Klee, V., Helly's theorem and its relatives, Proc. Sympos. Pure. Math., Vol. 7, AMS, Providence, R.I. 1963, pp. 101–108.Google Scholar
4. Erdös, P., On some problems of elementary and combinatorial geometry, Ann. Mat. Pura. Appl. (4) 103(1975), pp. 99–108.Google Scholar
5. Erdös, P. and Szekeres, G., A combinatorial problem in geometry, Compositio Math. 2 (1935), pp. 463–470.Google Scholar
6. Fabella, G. and O'Rourke, J., Twenty-two points with no empty hexagon (1986, manuscript).Google Scholar
7. Grünbaum, B., Convex poly topes, N.Y., 1967.Google Scholar
8. Harborth, H., Konvexe Funfecke in ebenen Punktmengen, Elem. Math. 33 (1978), pp. 116–118.Google Scholar
9. Horton, J.D., Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), pp. 482–484.Google Scholar
10. John, F., Extremum problems with inequalities as subsidiary conditions, Courant Ann. Volume , Interscience, N.Y., 1948, pp. 187–204.Google Scholar
11. Katchalski, M. and Meir, A., On empty triangles determined by points in the plane, Acta. Math. Hungar. (to appear).Google Scholar
12. McMullen, P., The maximum number of faces of a convex polytope, Mathematika 17 (1970), pp. 179–184.Google Scholar
13. Purdy, G.B., The minimum number of empty triangles, AMS Abstract 3 (1982), p. 318.Google Scholar