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On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space

Published online by Cambridge University Press:  01 March 2008

ADRIAN DUMITRESCU  and
CSABA D. TÓTH
ADRIAN DUMITRESCU
Affiliation:
Department of Computer Science, University of Wisconsin–Milwaukee, WI 53201-0784, USA (e-mail: ad@cs.uwm.edu)
CSABA D. TÓTH
Affiliation:
Department of Mathematics, University of Calgary, AB, CanadaT2N 1N4 (e-mail: cdtoth@ucalgary.ca)
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Abstract

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3-space, and in general in d dimensions.

  1. (i) The number of tetrahedra of minimum (non-zero) volume spanned by n points in 3 is at most , and there are point sets for which this number is . We also present an O(n3) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every , the maximum number of k-dimensional simplices of minimum (non-zero) volume spanned by n points in d is Θ(nk).

  2. (ii) The number of unit volume tetrahedra determined by n points in 3 is O(n7/2), and there are point sets for which this number is Ω(n3 log logn).

  3. (iii) For every , the minimum number of distinct volumes of all full-dimensional simplices determined by n points in d, not all on a hyperplane, is Θ(n).

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 2 , March 2008 , pp. 203 - 224
Copyright
Copyright © Cambridge University Press 2007

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