Mathematics

If Ω is the interior of a convex polygon in R 2 and f,g two asymptotic geodesics, we show that the distance function d(f(t),g(t)) is convex for t sufficiently large. The same result is obtained in the case ∂Ω is of class C 2 and the curvature of ∂Ω at the point f(∞)=g(∞) does not vanish. An example is provided for the necessity of the curvature assumption.

We show that, for vector spaces in which distance measurement is performed using a gauge, the existence of best coapproximations in 1 -codimensional closed linear subspaces implies in dimensions ?? that the gauge is a norm, and in dimensions ?? that the gauge is even a Hilbert space norm. We also show that coproximinality of all closed subspaces of a fixed dimension implies coproximinality of all subspaces of all lower finite dimensions.

After having investigated the packings derived by horo- and hyperballs related to simple frustum Coxeter orthoscheme tilings we consider the corresponding covering problems (briefly hyp-hor coverings) in n -dimensional hyperbolic spaces H n ( n=2,3 ). We construct in the 2− and 3− dimensional hyperbolic spaces hyp-hor coverings that are generated by simply truncated Coxeter orthocheme tilings and we determine their thinnest covering configurations and their densities. We prove that in the hyperbolic plane ( n=2 ) the density of the above thinnest hyp-hor covering arbitrarily approximate the universal lower bound of the hypercycle or horocycle covering density 12 √ π and in H 3 the optimal configuration belongs to the {7,3,6} Coxeter tiling with density ≈1.27297 that is less than the previously known famous horosphere covering density 1.280 due to L.~Fejes Tóth and K.~Böröczky. Moreover, we study the hyp-hor coverings in truncated orthosche\-mes {p,3,6} (6<p<7, p∈R) whose density function attains its minimum at parameter p≈6.45962 with density ≈1.26885 . That means that this locally optimal hyp-hor configuration provide smaller covering density than the former determined ≈1.27297 but this hyp-hor packing configuration can not be extended to the entirety of hyperbolic space H 3 .

Let K be a bounded convex domain in R 2 symmetric about the origin. The critical locus of K is defined to be the (non-empty compact) set of lattices Λ in R 2 of smallest possible covolume such that Λ∩K={0} . These are classical objects in geometry of numbers; yet all previously known examples of critical loci were either finite sets or finite unions of closed curves. In this paper we give a new construction which, in particular, furnishes examples of domains having critical locus of arbitrary Hausdorff dimension between 0 and 1 .

We prove the existence of the curvature measures for a class of U WDC sets, which is a direct generalization of U PR sets studied by Rataj and Zähle. Moreover, we provide a simple characterisation of U WDC sets in R 2 and prove that in R 2 the class of U WDC sets contains essentially all classes of sets known to admit curvature measures.

We represent minimal upper gradients of Newtonian functions, in the range 1?�p<??, by maximal directional derivatives along "generic" curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of p -weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along p -almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. The arising p -weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger's structure in the presence of a Poincaré inequality. It is moreover compatible with, and gives a geometric interpretation of, Gigli's abstract differentiable structure, whenever it exists. The p -weak charts give rise to a finite dimensional p -weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.

We prove that almost every triangle can be dissected only into n 2 triangles which have to be equal one another. Moreover, such a dissection is unique for every n . It turns out that to solve this "simple" problem it is convenient to use some tools from linear algebra, calculus and measure theory.

We prove the equivalence between the simplicial Orlicz cohomology and the Orlicz-de Rham cohomology in the case of Lie groups. Since the first one is a quasi-isometry invariant for uniformly contractible simplicial complexes with bounded geometry, we obtain the invariance of the second one in the case of contractible Lie groups. We also define the Orlicz cohomology of a Gromov-hyperbolic space relative to a point on its boundary at infinity, for which the same results are true.

This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given thickness.

A notion of differentiability is being proposed for maps between Wasserstein spaces of order 2 of smooth, connected and complete Riemannian manifolds. Due to the nature of the tangent space construction on Wasserstein spaces, we only give a global definition of differentiability, i.e. without a prior notion of pointwise differentiability. With our definition, however, we recover the expected properties of a differential. Special focus is being put on differentiability properties of pushforward maps induced by smooth maps between the underlying manifolds, and on convex mixing of differentiable maps, with an explicit construction of the differential.