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 $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$ is included in the convex hull of $U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles. Here we prove the existence of such a $j$ and $k$ for the more general case where $U_0$ and $U_1$ are compact sets in the plane such that $U_1$ is obtained from $U_0$ 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 $C^\infty$ 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 $C^\infty$ 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 $B$ [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

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

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 $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in\{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1−k}$ is included in the convex hull of $U_k\cup(\{A_0,A_1,A_2\}∖setminus\{A_j\})$. One could say disks instead of circles. Here we prove the existence of such a $j$ and $k$ for the more general case where $U_0$ and $U_1$ are compact sets in the plane such that $U_1$ is obtained from $U_0$ 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

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 $k \leq 3$ 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