Mathematics

Let X , Y be sets and let Φ , Ψ be mappings with domains X 2 and Y 2 respectively. We say that Φ and Ψ are combinatorially similar if there are bijections f:Φ( X 2 )→Ψ( Y 2 ) and g:Y→X such that Ψ(x,y)=f(Φ(g(x),g(y))) for all x , y∈Y . Conditions under which a given mapping is combinatorially similar to an ultrametric or a pseudoultrametric are found. Combinatorial characterizations are also obtained for poset-valued ultrametric distances recently defined by Priess-Crampe and Ribenboim.

We re-prove the classification of flexible octahedra, obtained by Bricard at the beginning of the XX century, by means of combinatorial objects satisfying some elementary rules. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic.

Similarly to the classic notion in E d , a subset of a positive diameter below π 2 of a hemisphere of the sphere S d is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies on S d . Our main theorem says that on S d complete bodies of diameter δ coincide with bodies of constant width δ .

All continuous translation invariant complex-valued valuations on Lebesgue measurable functions are completely classified. And all continuous rotation invariant complex-valued valuations on spherical Lebesgue measurable functions are also completely classified.

We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov's book. We obtain a complete answer for this question.

Brandolini et al. conjectured that all concrete lattice polytopes can multitile the space. We disprove this conjecture in a strong form, by constructing an infinite family of counterexamples in R 3 .

Given a compact connected set E in the unit disk B 2 , we give a new upper bound for the conformal capacity of the condenser ( B 2 ,E) in terms of the hyperbolic diameter t of E . Moreover, for t>0 we construct a set of diameter t and show by numerical computation that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is of constant hyperbolic width equal to t , the so called hyperbolic Reuleaux triangle.

We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without 2 -torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.

Kohlenbach and Leustean have shown in 2010 that any asymptotically nonexpansive self-mapping of a bounded nonempty UCW -hyperbolic space has a fixed point. In this paper, we adapt a construction due to Moloney in order to provide a sequence that converges strongly to such a fixed point.

Given n distinct points x 1 ,…, x n in R d , let K denote their convex hull, which we assume to be d -dimensional, and B=∂K its (d−1) -dimensional boundary. We construct an explicit one-parameter family of continuous maps f ε : S d−1 →K which, for ε>0 , are defined on the (d−1) -dimensional sphere and have the property that the images f ε ( S d−1 ) are codimension 1 submanifolds contained in the interior of K . Moreover, as the parameter ε goes to 0 + , the images f ε ( S d−1 ) converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B , appropriately defined. Several computer plots illustrating our results will be presented.