573 research outputs found
On -Gons and -Holes in Point Sets
We consider a variation of the classical Erd\H{o}s-Szekeres problems on the
existence and number of convex -gons and -holes (empty -gons) in a set
of points in the plane. Allowing the -gons to be non-convex, we show
bounds and structural results on maximizing and minimizing their numbers. Most
noteworthy, for any and sufficiently large , we give a quadratic lower
bound for the number of -holes, and show that this number is maximized by
sets in convex position
Compatible 4-Holes in Point Sets
Counting interior-disjoint empty convex polygons in a point set is a typical
Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let be a
set of points in the plane and in general position. A subset of ,
with four points, is called a -hole in if is in convex position and
its convex hull does not contain any point of in its interior. Two 4-holes
in are compatible if their interiors are disjoint. We show that
contains at least pairwise compatible 4-holes.
This improves the lower bound of which is implied by a
result of Sakai and Urrutia (2007).Comment: 17 page
Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems
This is an updated and expanded version of our earlier survey article
\cite{Gut5}. Section introduces the subject matter. Sections expose the basic material following the paradigm of elliptic, hyperbolic and
parabolic billiard dynamics. In section we report on the recent work
pertaining to the problems and conjectures exposed in the survey \cite{Gut5}.
Besides, in section we formulate a few additional problems and
conjectures. The bibliography has been updated and considerably expanded
Combinatorially two-orbit convex polytopes
Any convex polytope whose combinatorial automorphism group has two orbits on
the flags is isomorphic to one whose group of Euclidean symmetries has two
orbits on the flags (equivalently, to one whose automorphism group and symmetry
group coincide.) Hence, a combinatorially two-orbit convex polytope is
isomorphic to one of a known finite list, all of which are 3-dimensional: the
cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic
triacontahedron. The same is true of combinatorially two-orbit normal
face-to-face tilings by convex polytopes.Comment: 20 page
A better upper bound on the number of triangulations of a planar point set
We show that a point set of cardinality in the plane cannot be the vertex
set of more than straight-edge triangulations of its convex
hull. This improves the previous upper bound of .Comment: 6 pages, 1 figur
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