Related papers: Normal Cones and Thompson Metric

Let X be a linear space over K, K=R or K=C and let for n>1 \rho_i be s-convex semimodular defined on X for any i\in{1,...,n-1}. Put \rho=\max_{1\leq i \leq n-1}\{\rho_i\} and X_{\rho}= { x \in X: \rho(dx) < \infty for some d > 0 }. In this…

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree ($T$, $d$) is a metric space such that between any…

With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions…

In this article we treat two closely related problems: 1) the upper semi continuity property for Almgren minimal sets in regions with regular boundary, which guanrantees that the uniqueness property is well defined; and 2) the Almgren…

In this work, we look at the symmetry of normal modes in symmetric structures, particularly structures with cyclic symmetry. We show that normal modes of symmetric structures have different levels of symmetry, or symmetricity. One novel…

Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces:…

A unified expression for topological invariants has been proposed recently to describe the topological order in Dirac models belonging to any dimension and symmetry class. We uncover a correspondence between the curvature function that…

We show that the existence of constant scalar curvature K\"ahler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the…

Thurston introduced in his seminal work an asymmetric metric on Teichm\"uller space by the ratio of simple closed curve length. In this paper, we generalize the idea and define an asymmetric metric on the space of unit-area flat metrics…

If $K$ and $L$ are mutually dual closed convex cones in a Hilbert space with the metric projections onto them denoted by $P_K$ and $P_L$ respectively, then the following two assertions are equivalent: (i) $P_K$ is isotone with respect to…

We derive a sufficient condition guaranteeing that a singularly perturbed linear time-varying system is strongly monotone with respect to a matrix cone $C$ of rank $k$. This implies that the singularly perturbed system inherits the…

In this work we investigate the transfer of fundamental order and completeness properties between truncated Riesz spaces and their unitizations. Specifically, we provide characterizations and equivalences for several notions of…

We introduce a new type of norm for ordered vector spaces majorized by a proper (convex) cone that generalizes the notions of order unit norm and base norm. Then we give sufficient conditions to ensure its completeness. In the case of…

In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we…

In the Bondi-Sachs formulation of General Relativity space-time is foliated via a family of null cones. If these null cones are defined such that their vertices are traced by a regular world-line then the metric tensor has to obey…

In this paper, we introduce the $\mathcal{F}$-metric space concept, which generalizes the metric space notion. We define a natural topology $\tau_{\mathcal{F}}$ in such spaces and we study their topological properties. Moreover, we…

We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological condition) are contained in a quasisymmetric arc. This…

We study singular K\"ahler-Einstein metrics that are obtained as non-collapsed limits of polarized K\"ahler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the…

For a proper cone $K$ and its dual cone $K^*$ in $\mathbb R^n$, the complementarity set of $K$ is defined as ${\mathbb C}(K)=\{(x,y): x\in K,\; y\in K^*,\, x^\top y=0\}$. It is known that ${\mathbb C}(K)$ is an $n$-dimensional manifold in…

A universal topological marker has been proposed recently to map the topological invariants of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the…