We study the structure of Lipschitz and Hölder-type spaces and their preduals on general metric spaces, and give applications to the uniform structure of Banach spaces. In particular we resolve a problem of Weaver who asks whether if M is a compact metric space and 0 < α < 1, it is always true the space of Hölder continuous functions of class α is isomorphic to `∞. We show that, on the contrary...

We prove that a closed, geodesically convex subset C of P 2 (R) is closed with respect to weak convergence in P 2 (R). This means that if (μn) ⊂ C is such that μn ⇀ μ in duality with continuous bounded functions and supn ∫ |x|dμn <∞, then μ ∈ C as well.

Developed in 1999 by Akemann, Anderson, and Weaver, the spectral scale of an n × n matrix A, is a convex, compact subset of R that reveals important spectral information about A [4]. In this paper we present new information found in the spectral scale of a matrix. Given a matrix A = A1 + iA2 with A1 and A2 self-adjoint and A2 6= 0, we show that faces in the boundary of the spectral scale of A t...

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P . We construct a set of n points in general position in the plane with only ≈ 1.62n empty triangles, ≈ 1.94n empty quadrilaterals, ≈ 1.02n empty pentagons, and ≈ 0.2n empty hexagons.

Let D be a nonempty and convex subset of a Banach spaces E. The set D is called proximinal if for each x∈E, there exists an element y∈D such that ||x-y|| = d(x,D), where d(x,D) = inf{||x-z||: z∈D}. Let CB(D), CCB(D), K(D) and P(D) denote the families of nonempty closed bounded subsets, nonempty closed convex bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of D...

We give an alternative proof to the well known fact that each convex compact centrally symmetric subset of R2 containing the origin is a zonoid, i.e., the range of a two dimensional vector measure, and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a Chebyshev measure. We give a condition under which a two dimension...

We establish that every bounded convex domain in C with an automorphism orbit accumulation at a boundary point at which the domain has a sphere contact from inside admits a non-compact 1-parameter subgroup of automorphisms. Notice that this in particular implies that no Teichmüller domain of a Riemann surface of genus g > 1 can be holomorphically imbedded as a convex domain in C .

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.

insisting that the equality sign holds when k = n. Here, x[X] > • • • > x[n] are the xt arranged in decreasing order and, similarly, y[X] > • • • > y[n]. If (1) is only required for the increasing (decreasing) convex functions on R then one speaks of weak sub-majorization x >, respectively). The first is equivalent to (2). Let & be an open convex subset of R...