Mathematics

In this article we provide a definition of the concept of angle space. Unlike previous works on this topic, we do it for sets with a notion of betweenness that are not necessarily metric spaces. After this, we solve two problems proposed by K. Menger in the Annals of Mathematics in 1931. The first one consists in characterizing angle spaces that can be embedded in the euclidean plane E 2 . We answer this question in a general way characterizing those that can be embedded in E n . And the second one concerns a characterization of those angle spaces admitting a distance function compatible with the angle space structure.

Let x and y be two unit vectors in a normed plane R 2 . We say that x is Birkhoff orthogonal to y if the line through x in the direction y supports the unit disc. A B-measure (Fankhänel 2011) is an angular measure μ on the unit circle for which μ(C)=π/2 whenever C is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure.

The aim of this note is twofold: to give a short proof of the results in [S. Larson, A bound for the perimeter of inner parallel bodies, J. Funct. Anal. 271 (2016), 610-619] and [G. Domokos and Z. Lángi, The isoperimetric quotient of a convex body decreases monotonically under the eikonal abrasion model, Mathematika 65 (2019), 119-129]; and to generalize them to the anisotropic case.

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, because its potential is discontinuous, while the standard theory of Gibbs measures requires continuous (actuallly, Hölder) potentials. In this paper, we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar Dirichlet forms on fractals. Moreover, we shall see that Kusuoka's measure can be recovered in a simple way from the matrix-valued Gibbs measure.

We find a formula for the area of disks tangent to a given disk in an Apollonian disk packing (corona) in terms of a certain novel arithmetic Zeta function. The idea is based on "tangency spinors" defined for pairs of tangent disks.

The depth function of three numbers representing curvatures of three mutually tangent circles is introduced. Its 2D plot leads to a partition of the moduli space of the triples of mutually tangent circles/disks that is unexpectedly a beautiful fractal, the general form of which resembles that of an Apollonian disk packing, except that it consists of ellipses instead of circles.

The configuration space of tricycles (triples of disks in contact) is shown to coincide with the complex plane resulting as a projective space costructed from the tangency and Pauli spinors. Remarkably, the fractal of the depth functions assumes a particularly simple and elegant form. Moreover, the factor space due to a certain symmetry group provides a parametrization of the Apollonian disk packings.

In the present paper we study $\SXR$ and $\HXR$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the "surface of a geodesic triangle". Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\SXR$ and $\HXR$ tetrahedron. Moreover, we generalize the famous Menelaus' and Ceva's theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\SXR$ and $\HXR$ geometries described by E. Molnár in \cite{M97}.

We show that any bounded domain in a doubling quasiconvex metric space can be approximated from inside and outside by uniform domains.

We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is computed with respect to an arbitrary homogeneous distance. Among the arguments of the proof, we point out the differentiability properties of intrinsic graphs and a chain rule for intrinsic differentiable functions.