Zone diagrams in compact subsets of uniformly convex normed spaces

A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matoušek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space, provided that either the sites or the convex subset satisfy a certain mild condition. The proof is based on the Schauder fixed point theorem, the Curtis-Schori theorem regarding the Hilbert cube, and on recent results concerning the characterization of Voronoi cells as a collection of line segments and their geometric stability with respect to small changes of the corresponding sites. Along the way we obtain the continuity of the Dom mapping as well as interesting and apparently new properties of Voronoi cells.     

Dimension of compact metric spaces

We give a survey of old and new results in dimension theory of compact metric spaces. Most of the relatively new results presented in the survey are based on the cohomological dimension approach. We complement the survey by stating the basics of cohomological dimension theory and listing some of its applications beyond the dimension theory.     

Covering compact metric spaces greedily

A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvátal’s analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of the n-dimensional sphere and n-dimensional Euclidean space to give short and transparent proofs of several best known bounds obtained from constructions in the literature on sphere coverings.     

Generic properties of compact metric spaces

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose elements enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.     

Weak-open compact images of metric spaces

The main results of this paper are that (1) a space X is g-developable if and only if it is a weak-open π image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.     

Hermitian metric rigidity on compact foliated manifolds

We prove a Hermitian metric rigidity theorem for leafwise symmetric Kaehler metrics on compact manifolds with smooth foliations. This provides applications to the study of the geometry of foliations as well as Kaehler manifolds that contain some symmetric geometry.     

Images of locally compact metric spaces

We establish relationships between locally compact-metric spaces and all kinds of spaces with mk-system (compact k-network) by means of compact-covering mappings, quotient mapping and closed mappings. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multi-valued non-self-mappings on convex metric spaces

In this paper we prove common fixed-point theorems for a pair of multi-valued non-self-mappings in metrically convex metric spaces. Our results generalize and extend both theorems of Itoh, (Comment. Math. Univ. Carolin. 18 (1977) 247) and Khan, (Pacific J. Math. 95 (1981) 337), the theorem of Assad, (Boll. Un. Math. Ital. 4 (1973) 1), the theorem of Rhoades, (Comment. Math. Univ. Carolin. 37 (1996) 401) and the main theorem of Assad and Kirk, (Pacific J. Math. 43 (1972) 535).     

Holomorphic manifolds on locally convex spaces

Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.     

Expanding maps on compact metric spaces

The notions of locally expansive, positively expansive, expanding in the sense of Ruelle and expanding in the sense of Duvall and Husch are equivalent in a quite general setting.     

On hilbertian subsets of finite metric spaces

The following result is proved: For everyε>0 there is aC(ε)>0 such that every finite metric space (X, d) contains a subsetY such that |Y|≧C(ε)log|X| and (Y, d _Y) embeds (1 +ε)-isomorphically into the Hilbert spacel ₂. The authors are grateful to Haim Wolfson for some discussions related to the content of this paper.     

Open subspaces of locally compact metric spaces

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.     

Strong consistency of nonparametric Bayes density estimation on compact metric spaces with applications to specific manifolds

This article considers a broad class of kernel mixture density models on compact metric spaces and manifolds. Following a Bayesian approach with a nonparametric prior on the location mixing distribution, sufficient conditions are obtained on the kernel, prior and the underlying space for strong posterior consistency at any continuous density. The prior is also allowed to depend on the sample size n and sufficient conditions are obtained for weak and strong consistency. These conditions are verified on compact Euclidean spaces using multivariate Gaussian kernels, on the hypersphere using a von Mises-Fisher kernel and on the planar shape space using complex Watson kernels.     

Positively expansive homeomorphisms on compact metric spaces

We give a new and simple proof to show that no homeomorphism of infinite compact metric spaces is positively expansive.     

Covering maps of spaces of compact subsets

Mappings generated by covering maps of metric spaces are considered in the spaces of compact subsets of metric spaces. Sufficient conditions for the covering property of maps of spaces of compact sets and sufficient conditions for the existence of coincidence sets of maps of metric spaces are obtained.