54 research outputs found
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
K. Adaricheva and M. Bolat have recently proved that if and are
circles in a triangle with vertices , then there exist and such that is included in the convex hull
of . One could say disks instead of
circles. Here we prove the existence of such a and for the more general
case where and are compact sets in the plane such that is
obtained from by a positive homothety or by a translation. Also, we give
a short survey to show how lattice theoretical antecedents, including a series
of papers on planar semimodular lattices by G. Gratzer and E. Knapp, lead to
our result.Comment: 28 pages, 7 figure
Straight projective-metric spaces with centers
It is proved that a straight projective-metric space has an open set of
centers, if and only if it is either the hyperbolic or a Minkowskian geometry.
It is also shown that if a straight projective-metric space has some finitely
many well-placed centers, then it is either the hyperbolic or a Minkowskian
geometry.Comment: 11 page
Finding Needles in a Haystack
Convex polygons are distinguishable among the piecewise convex domains by comparing their visual angle functions on any surrounding circle. This is a consequence of our main result, that every segment in a multi\-curve can be reconstructed from the masking function of the multicurve given on any surrounding circle
A characterization of the Radon transform and its dual on Euclidean space
In this paper we present a characterization of the Radon transform without any restriction on its range. We also consider the boomerang transform [5], which is essentially the dual of the Radon transform
New unified Radon inversion formulas
We prove two unified Radon inversion formulas using elementary geometry and analysis
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
K. Adaricheva and M. Bolat have recently proved that if and are circles in a triangle with vertices , then there exist and such that is included in the convex hull of . One could say disks instead of circles. Here we prove the existence of such a and for the more general case where and are compact sets in the plane such that is obtained from by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Gr\"atzer and E. Knapp, lead to our result
Euler’s ratio-sum formula in projective-metric spaces
We prove that Euler’s ratio-sum formula is valid in a projective-metric space if and only if it is either elliptic, hyperbolic, or Minkowskian
Tiling a circular disc with congruent pieces
In this note, we prove that any monohedral tiling of the closed circular unit disc with k≤3 topological discs as tiles has a k-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer, and Guy in 1994
Tiling a circular disc with congruent pieces
In this note we prove that any monohedral tiling of the closed circular unit disc with topological discs as tiles has a -fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer and Guy in 1994
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