61 research outputs found
On the number of non-hexagons in a planar tiling
We give a simple proof of T. Stehling's result, that in any normal tiling of
the plane with convex polygons with number of sides not less than six, all
tiles except the finite number are hexagons.Comment: 2 pages, 2 figure
On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape
We study the lengths of curves passing through a fixed number of points on
the boundary of a convex shape in the plane. We show that for any convex shape
, there exist four points on the boundary of such that the length of any
curve passing through these points is at least half of the perimeter of . It
is also shown that the same statement does not remain valid with the additional
constraint that the points are extreme points of . Moreover, the factor
cannot be achieved with any fixed number of extreme points. We
conclude the paper with few other inequalities related to the perimeter of a
convex shape.Comment: 7 pages, 8 figure
Long geodesics on convex surfaces
We review the theory of intrinsic geometry of convex surfaces in the
Euclidean space and prove the following theorem: if the surface of a convex
body K contains arbitrary long closed simple geodesics, then K is an isosceles
tetrahedron.Comment: 8 pages, 10 figure
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
Any cyclic quadrilateral can be inscribed in any closed convex smooth curve
We prove that any cyclic quadrilateral can be inscribed in any closed convex
-curve. The smoothness condition is not required if the quadrilateral is a
rectangle.Comment: 6 pages, 10 figure
- …