The setting for this paper is n-dimensional Euclidean space Rn. Let n denote the set of convex bodies (compact, convex subsets with nonempty interiors) and n o denote the subset of n that consists of convex bodies with the origin in their interiors. Denote by voli(K | ξ) the i-dimensional volume of the orthogonal projection of K onto an idimensional subspace ξ ⊂Rn. Affine quermassintegrals are ...

Let M be the collection of all intersections of balls, considered as a subset of the hyperspace H of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. We prove that M is uniformly very porous if and only if the space fails the Mazur intersection property.

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 lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Ω) = {X ∩Ω | X ∈ Co(V )}, for some f...

In this paper, the notions of L-fuzzy subalgebra degree and L-subalgebras on an effect algebra are introduced some characterizations given. We use four kinds cut sets L-subsets to characterize degree. induce convexity by degree, we prove that a morphism between two algebras is preserving mapping monomorphism convex-to-convex mapping. Finally, it proved set all can form L-convexity, its L-convex...

The main results of the paper: (1) The dual Banach space X∗ contains a linear subspace A ⊂ X∗ such that the set A of all limits of weak∗ convergent bounded nets in A is a proper norm-dense subset of X∗ if and only if X is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. (2) Let X be a non-reflexive Banach space. Then there exists a convex subse...

In this paper, we study the convergence of the sequence defined by x0 ∈ C, xn+1 = αnx + (1− αn)Txn, n = 0, 1, 2, . . . , where 0 ≤ αn ≤ 1 and T is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

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.