Related papers: Saturating Constructions for Normed Spaces II

For a left vector space V over a totally ordered division ring F, let Co(V) denote the lattice of convex subsets of V. We prove that every lattice L can be embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite…

Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed…

It is proved that for an h-homogeneous space X the following conditions are equivalent: 1) X is a densely homogeneous space with a dense complete subspace; 2) X is $\sigma$-discretely controlled.

Each relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. Suppose that X is infinite, connected and of bounded degree. A first-order sentence in the language of X is almost surely true…

For any given compact C^2 hypersurface \Sigma in {\bf R}^{2n} bounding a strictly convex set with nonempty interior, in this paper an invariant \varrho_n(\Sigma) is defined and satisfies \varrho_n(\Sigma)\ge [n/2]+1, where [a] denotes the…

An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable…

A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube. We prove lower…

For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $\Gamma(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$,…

we prove that if $X$ is a locally compact $\sigma$-compact space then on its quotient, $\gamma(X)$ say, determined by the algebra of all real valued bounded continuous functions on $X$, the quotient topology and the completely regular…

The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…

A closed real subspace V of a complex Hilbert space H is called standard if V intersects iV trivially and and V + i V is dense in H. In this note we study several aspects of the geometry of the space Stand(H) of standard subspaces. In…

The paper studies a general norm minimization problem on a product of normed vector spaces. We establish dual necessary and sufficient optimality conditions and derive explicit formulas for the corresponding solution sets. These formulas…

We prove that if $\Om \subseteq \R^2$ is bounded and $\R^2 \setminus \Om$ satisfies suitable structural assumptions (for example it has a countable number of connected components), then $W^{1,2}(\Om)$ is dense in $W^{1,p}(\Om)$ for every…

This paper forms the third part of a series by the authors [GMV1,GMV2] concerning the structure theory of nilspaces. A nilspace is a compact space $X$ together with closed collections of cubes $C^n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$,…

The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous…

Given a compact pseudo-metric space, we associate to it upper and lower dimensions, depending only on the metric. Then we construct a doubling metric for which the measure of a dillated ball is closely related to these dimensions.

Let $X$ be a finite-dimensional normed space and let $Y \subseteq X$ be its proper linear subspace. The set of all minimal projections from $X$ to $Y$ is a convex subset of the space all linear operators from $X$ to $X$ and we can consider…

We give a new proof of a theorem of Loos stating that a Riemannian symmetric space X with rectangular unit lattice is a symmetric R-space. For this we construct explicitly an isometric extrinsically symmetric embedding of X in a Euclidean…

Decomposition spaces are a class of function spaces constructed out of well-behaved coverings and partitions of unity of a set. The structure of the covering of the set determines the properties of the decomposition space. Besov spaces,…

We use non-perturbative U-duality symmetries of type II strings to construct new vacuum solutions. In some ways this generalizes the F-theory vacuum constructions. We find the possibilities of new vacuum constructions are very limited.…