96 research outputs found
Decompositions of a polygon into centrally symmetric pieces
In this paper we deal with edge-to-edge, irreducible decompositions of a
centrally symmetric convex -gon into centrally symmetric convex pieces.
We prove an upper bound on the number of these decompositions for any value of
, and characterize them for octagons.Comment: 17 pages, 17 figure
The robustness of equilibria on convex solids
We examine the minimal magnitude of perturbations necessary to change the
number of static equilibrium points of a convex solid . We call the
normalized volume of the minimally necessary truncation robustness and we seek
shapes with maximal robustness for fixed values of . While the upward
robustness (referring to the increase of ) of smooth, homogeneous convex
solids is known to be zero, little is known about their downward robustness.
The difficulty of the latter problem is related to the coupling (via integrals)
between the geometry of the hull \bd K and the location of the center of
gravity . Here we first investigate two simpler, decoupled problems by
examining truncations of \bd K with fixed, and displacements of with
\bd K fixed, leading to the concept of external \rm and internal \rm
robustness, respectively. In dimension 2, we find that for any fixed number
, the convex solids with both maximal external and maximal internal
robustness are regular -gons. Based on this result we conjecture that
regular polygons have maximal downward robustness also in the original, coupled
problem. We also show that in the decoupled problems, 3-dimensional regular
polyhedra have maximal internal robustness, however, only under additional
constraints. Finally, we prove results for the full problem in case of 3
dimensional solids. These results appear to explain why monostatic pebbles
(with either one stable, or one unstable point of equilibrium) are found so
rarely in Nature.Comment: 20 pages, 6 figure
On the average number of normals through points of a convex body
In 1944, Santal\'o asked about the average number of normals through a point
of a given convex body. Since then, numerous results appeared in the literature
about this problem. The aim of this paper is to give a concise summary of these
results, with some new, recent developments. We point out connections of this
problem to static equilibria of rigid bodies as well as to geometric partial
differential equations of surface evolution.Comment: 15 page
On the Hadwiger numbers of starlike disks
The Hadwiger number of a topological disk in is the
maximal number of pairwise nonoverlapping translates of that touch . It
is well known that for a convex disk, this number is six or eight. A conjecture
of A. Bezdek., K. and W. Kuperberg says that the Hadwiger number of a starlike
disk is at most eight. A. Bezdek proved that this number is at most seventy
five for any starlike disk. In this note, we prove that the Hadwiger number of
a starlike disk is at most thirty five. Furthermore, we show that the Hadwiger
number of a topological disk such that (\conv J) \setminus J is
connected, is six or eight.Comment: 13 pages, 8 figure
On the perimeters of simple polygons contained in a disk
A simple -gon is a polygon with edges with each vertex belonging to
exactly two edges and every other point belonging to at most one edge. Brass
asked the following question: For odd, what is the maximum perimeter
of a simple -gon contained in a Euclidean unit disk?
In 2009, Audet, Hansen and Messine answered this question, and showed that
the optimal configuration is an isosceles triangle with a multiple edge,
inscribed in the disk. In this note we give a shorter and simpler proof of
their result, which we generalize also for hyperbolic disks, and for spherical
disks of sufficiently small radii.Comment: 6 pages, 2 figure
On the volume of the convex hull of two convex bodies
In this note we examine the volume of the convex hull of two congruent copies
of a convex body in Euclidean -space, under some subsets of the isometry
group of the space. We prove inequalities for this volume if the two bodies are
translates, or reflected copies of each other about a common point or a
hyperplane containing it. In particular, we give a proof of a related
conjecture of Rogers and Shephard.Comment: 9 pages, 3 figure
Ellipsoid characterization theorems
In this note we prove two ellipsoid characterization theorems. The first one is that if K is a convex body in a normed space with unit ball M, and for any point p ∉ K and in any 2-dimensional plane P intersecting intK and containing p, there are two tangent segments of the same normed length from p to K, then K and M are homothetic ellipsoids. Furthermore, we show that if M is the unit ball of a strictly convex, smooth norm, and in this norm billiard angular bisectors coincide with Busemann angular bisectors or Glogovskij angular bisectors, then M is an ellipse
On the perimeters of simple polygons contained in a plane convex body
A simple n-gon is a polygon with n edges such that each vertex belongs to
exactly two edges and every other point belongs to at most one edge. Brass,
Moser and Pach asked the following question: For n > 3 odd, what is the maximum
perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet,
Hansen and Messine answered this question, and showed that the supremum is the
perimeter of an isosceles triangle inscribed in the disk, with an edge of
multiplicity n-2. L\'angi generalized their result for polygons contained in a
hyperbolic disk. In this note we find the supremum of the perimeters of simple
n-gons contained in an arbitrary plane convex body in the Euclidean or in the
hyperbolic plane.Comment: 7 pages, 7 figure
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