43 research outputs found
Billiards in convex bodies with acute angles
In this paper we investigate the existence of closed billiard trajectories in
not necessarily smooth convex bodies. In particular, we show that if a body
has the property that the tangent cone of every
non-smooth point is acute (in a certain sense) then there is
a closed billiard trajectory in .Comment: 8 pages, 2 figure
On the Circle Covering Theorem by A. W. Goodman and R. E. Goodman
In 1945, A. W. Goodman and R. E. Goodman proved the following conjecture by
P. Erd\H{o}s: Given a family of (round) disks of radii , ,
in the plane it is always possible to cover them by a disk of radius , provided they cannot be separated into two subfamilies by a straight line
disjoint from the disks. In this note we show that essentially the same idea
may work for different analogues and generalizations of their result. In
particular, we prove the following: Given a family of positive homothetic
copies of a fixed convex body with homothety
coefficients it is always possible to cover them
by a translate of , provided they
cannot be separated into two subfamilies by a hyperplane disjoint from the
homothets.Comment: 7 pages, 3 figure
Flip cycles in plabic graphs
Planar bicolored (plabic) graphs are combinatorial objects introduced by
Postnikov to give parameterizations of the positroid cells of the totally
nonnegative Grassmannian . Any two plabic graphs for
the same positroid cell can be related by a sequence of certain moves. The flip
graph has plabic graphs as vertices and has edges connecting the plabic graphs
which are related by a single move. A recent result of Galashin shows that
plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic
zonotope . Taking this perspective, we show that the fundamental group
of the flip graph is generated by cycles of length 4, 5, and 10, and use this
result to prove a related conjecture of Dylan Thurston about triple crossing
diagrams. We also apply our result to make progress on an instance of the
generalized Baues problem.Comment: 26 pages, 7 figures. Journal versio
Elementary approach to closed billiard trajectories in asymmetric normed spaces
We apply the technique of K\'aroly Bezdek and Daniel Bezdek to study billiard
trajectories in convex bodies, when the length is measured with a (possibly
asymmetric) norm. We prove a lower bound for the length of the shortest closed
billiard trajectory, related to the non-symmetric Mahler problem. With this
technique we are able to give short and elementary proofs to some known
results.Comment: 10 figures added. The title change