5,544 research outputs found
Bases in Systems of Simplices and Chambers
We consider a finite set of points in the -dimensional affine space
and two sets of objects that are generated by the set : the system
of -dimensional simplices with vertices in and the system of
chambers. The incidence matrix ,
, , induces the notion of linear
independence among simplices (and among chambers). We present an algorithm of
construction of bases of simplices (and bases of chambers). For the case
such an algorithm was described in the author's paper {\em Combinatorial bases
in systems of simplices and chambers} (Discrete Mathematics 157 (1996) 15--37).
However, the case of -dimensional space required a different technique. It
is also proved that the constructed bases of simplices are geometrical
There is no tame triangulation of the infinite real Grassmannian
We show that there is no triangulation of the infinite real Grassmannian of
k-planes in R^\infty which is nicely situated with respect to the coordinate
axes. In terms of matroid theory, this says there is no triangulation of the
Grassmannian subdividing the matroid stratification. This is proved by an
argument in projective geometry, considering a specific sequence of
configurations of points in the plane.Comment: 11 page
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
We formulate and give partial answers to several combinatorial problems on
volumes of simplices determined by points in 3-space, and in general in
dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by
points in \RR^3 is at most , and there are point sets
for which this number is . We also present an time
algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby
extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for
every k,d\in \NN, , the maximum number of -dimensional
simplices of minimum (nonzero) volume spanned by points in \RR^d is
. (ii) The number of unit-volume tetrahedra determined by
points in \RR^3 is , and there are point sets for which this
number is . (iii) For every d\in \NN, the minimum
number of distinct volumes of all full-dimensional simplices determined by
points in \RR^d, not all on a hyperplane, is .Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings
of the ACM-SIAM Symposium on Discrete Algorithms, 200
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