9 research outputs found
The number of unit-area triangles in the plane: Theme and variations
We show that the number of unit-area triangles determined by a set of
points in the plane is , improving the earlier bound
of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two
special cases of this problem: (i) We show, using a somewhat subtle
construction, that if consists of points on three lines, the number of
unit-area triangles that spans can be , for any triple of
lines (it is always in this case). (ii) We show that if is a {\em
convex grid} of the form , where , are {\em convex} sets of
real numbers each (i.e., the sequences of differences of consecutive
elements of and of are both strictly increasing), then determines
unit-area triangles
Maximal Area Triangles in a Convex Polygon
The widely known linear time algorithm for computing the maximum area
triangle in a convex polygon was found incorrect recently by Keikha et.
al.(arXiv:1705.11035). We present an alternative algorithm in this paper.
Comparing to the only previously known correct solution, ours is much simpler
and more efficient. More importantly, our new approach is powerful in solving
related problems
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
Multilinear generalized Radon transforms and point configurations
We study multilinear generalized Radon transforms using a graph-theoretic
paradigm that includes the widely studied linear case. These provide a general
mechanism to study Falconer-type problems involving -point
configurations in geometric measure theory, with , including the
distribution of simplices, volumes and angles determined by the points of
fractal subsets , . If denotes the set
of noncongruent -point configurations determined by , we show that if
the Hausdorff dimension of is greater than , then the
-dimensional Lebesgue measure of is positive. This
compliments previous work on the Falconer conjecture (\cite{Erd05} and the
references there), as well as work on finite point configurations
\cite{EHI11,GI10}. We also give applications to Erd\"os-type problems in
discrete geometry and a fractal regular value theorem, providing a multilinear
framework for the results in \cite{EIT11}.Comment: 27 pages, no figures. To appear, Forum Mathematicu
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
The Number of Unit-Area Triangles in the Plane: Theme and Variations *
Abstract We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n 20/9 ), improving the earlier bound O(n 9/4 ) of Apfelbaum and Sharir (ii) We show that if S is a convex grid of the form A × B, where A, B are convex sets of n 1/2 real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n 31/14 ) unit-area triangles
Extremal problems on triangle areas in two and three dimensions
The study of extremal problems on triangle areas was initiated in a series of papers by Erdős and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles. In the plane, our main result is an O(n 44/19) = O(n 2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n 7/3) from 1992. We also make progress in a number of important special cases: We show that (i) For points in convex position, there exist n-element point sets that span Ω(n log n) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most 2 3 (n2 −n); there exist n-element point sets (for arbitrarily large n) that span (6/π 2 − o(1))n 2 minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point set