Eine Charakterisierung der Sphäre im E3

It is well known, that in E³ the spheres are the closed convex C² -surfaces having the property, that each of their closed C²-curves with the total geodesic curvature O bisects the area of the surface. This characterization will be transmitted into the theory of convex surfaces founded by A.D.Alexandrov, where convex surfaces without any differentiability property are studied.     

On the generalized Stepanov theorem

The generalized Stepanov theorem is derived from the Alexandrov theorem on the twice differentiability of convex functions. A parabolic version of the generalized Stepanov theorem is also proved.

     

Vector measures and strict topologies

Let X be a completely regular Hausdorff space and Cb(X) be the space of all real-valued bounded continuous functions on X, endowed with the strict topology βσ. We study topological properties of continuous and weakly compact operators from Cb(X) to a locally convex Hausdorff space in terms of their representing vector measures. In particular, Alexandrov representation type theorems are derived. Moreover, a Yosida-Hewitt type decomposition for weakly compact operators on Cb(X) is given.     

Infinite curvature on typical convex surfaces

Solving a long-standing open question of Zamfirescu, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of Alexandrov spaces of bounded curvature, and show continuity properties for this notion.     

拓扑线性空间中的Drop定理与Drop性质

给出了拓扑线性空间中的一个Drop定理.利用此Drop定理,证明了拓扑线性空间中的每个序列紧凸集具有Drop性质;每个可数紧闭凸集具有拟Drop性质.而且结出了拓扑线性空间中Drop性质和拟Drop性质的序列流特征.也讨论了Drop性质和拟Drop性质与泛函取极值之间的联系.     

A conjecture on convex polyhedra

We select a class of pyramids of a particular shape and propose a conjecture that precisely these pyramids are of greatest surface area among the closed convex polyhedra having evenly many vertices and the unit geodesic diameter. We describe the geometry of these pyramids. The confirmation of our conjecture will solve the “doubly covered disk” problem of Alexandrov. Through a connection with Reuleaux polygons we prove that on the plane the convex n-gon of unit diameter, for odd n, has greatest area when it is regular, whereas this is not so for even n.     

Isotopy-invariant topological measures on closed orientable surfaces of higher genus

Given a closed orientable surface Σ of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on Σ and the convex compact set of additive functions on the set of isotopy classes of certain subsurfaces of Σ. We then construct such additive functions, and thus isotopy-invariant topological measures, from probability measures on Σ together with some additional data. The map associating topological measures to probability measures is affine and continuous. Certain Dirac measures map to simple topological measures, while the topological measures due to Py and Rosenberg arise from the normalized Euler characteristic.     

Cohomogeneity one Alexandrov spaces

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T ⁿ⁻¹ action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of \mathbbRP² \mathbb{R}{P^2} , which is not a manifold.     

Integration of ε-Fenchel Subdifferentials and Maximal Cyclic Monotonicity

This paper concerns the integration of ε-Fenchel subdifferentials of proper lower semicontinuous convex functions defined on arbitrary topological vector spaces. We make use of integration tools to provide a representation formula of the approximate subdifferential of convex functions, and also to identify the class of maximal cyclically monotone families of operators.     

Warped products of Hadamard spaces

In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved by warped products with convex warping functions. Received: 9 January 1998/ Revised version: 12 March 1998     

Quasi-convex subsets in Alexandrov spaces with lower curvature bound

We introduce quasi-convex subsets in Alexandrov spaces with lower curvature bound, which include not only all closed convex subsets without boundary but also all extremal subsets. Moreover, we explore several essential properties of such kind of subsets including a generalized Liberman theorem. It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alexandrov spaces with lower curvature bound.     

Dimensions of ℝ-trees and self-similar fractal spaces of nonpositive curvature

We study various dimensions of spaces with nonpositive curvature in the A. D. Alexandrov sense, in particular, of ?-trees. We find some conditions necessary and sufficient for the metric space to be an ?-tree and clarify relations between the topological, Hausdorff, entropy, and rough dimensions. We build the examples of ?-trees and CAT(0)-spaces in which strict inequalities between the topological, Hausdorff, and entropy dimensions hold; we also show that the Hausdorff and entropy dimensions can be arbitrarily large while the topological dimension remains fixed.     

Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds

Mathematische Zeitschrift - We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in...     

DIFFERENTIABILITY OF CONVEX FUNCTIONS ON SUBLINEAR TOPOLOGICAL SPACES AND VARIATIONAL PRINCIPLES IN LOCALLY CONVEX SPACES

This paper presents a type of variational principles for real valued w lower semicon-tinuous functions on certain subsets in duals of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.     

Carathéodory-Fejér interpolation in locally convex topological vector spaces

We study Carathéodory-Herglotz functions whose values are continuous operators from a locally convex topological vector space which admits the factorization property into its conjugate dual space. We show how this case can be reduced to the case of functions whose values are bounded operators from a Hilbert space into itself.     

Convex conjugates of integral functionals

We consider a convex integral functional on a functional space V andcompute its greatest extension to the algebraic bidual space V^**, among all convex functions which are lower semicontinuous with respect tothe *-weak topology o(V^** ; V^*).Such computations are usually performed to extend these functionals to sometopological closures. In the present paper, no a priori topological restrictionsare imposed on the extended domain. As a consequence, this extended functionalis a valuable first step for the computation of the exact shape of the minimizersof the conjugate convex integral functional subject to a convex constraint,in full generality: without constraint qualification. These convex integralfunctionals are sometimes called entropies, divergences or energies. Our proofsmainly rely on basic convex duality and duality in Orlicz spaces.     

A characterization of hemispheres

We prove that for constant contact angle γ=0, a capillary surface over a convex domain has no umbilical points unless that the surface is a hemisphere. The method involves the comparison of a lower hemisphere with the given surface at a second-ordered contact point and it is based on an argument of Alexandrov.     

Fuchsian convex bodies: basics of Brunn–Minkowski theory

The hyperbolic space ${\mathbb{H}^d}$ can be defined as a pseudo-sphere in the (d + 1) Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that ${\mathbb{H}^d/\Gamma}$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov–Fenchel and Brunn–Minkowski inequalities. Here the inequalities are reversed.     

On integration with respect to LCTVS-valued finitely additive measures

In this paper we develop an integration theory for scalar functions with respect to finitely additive measures taking values in a locally convex topological vector space, using both a weak and a strong approach. We also compare these kinds of integration with the more general Burkill-Cesari theory.     

An Alexandrov–Fenchel-type inequality for hypersurfaces in the sphere

We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov–Fenchel-type inequality for strictly convex hypersurfaces in the n-dimensional sphere, \(n \ge 3\).