A new survey: Cone metric spaces

The purpose of this new survey paper is, among other things, to collect in one place most of the articles on cone (abstract, K-metric) spaces, published after 2007. This list can be useful to young researchers trying to work in this part of functional and nonlinear analysis. On the other hand, the existing review papers on cone metric spaces are updated. The main contribution is the observation that it is usually redundant to treat the case when the underlying cone is solid and non-normal. Namely, using simple properties of cones and Minkowski functionals, it is shown that the problems can be usually reduced to the case when the cone is normal, even with the respective norm being monotone. Thus, we offer a synthesis of the respective fixed point problems arriving at the conclusion that they can be reduced to their standard metric counterparts. However, this does not mean that the whole theory of cone metric spaces is redundant, since some of the problems remain which cannot be treated in this way, which is also shown in the present article.Comment: 27 page

Intersection Theorems for Closed Convex Sets and Applications

A number of landmark existence theorems of nonlinear functional analysis follow in a simple and direct way from the basic separation of convex closed sets in finite dimension via elementary versions of the Knaster-Kuratowski-Mazurkiewicz principle - which we extend to arbitrary topological vector spaces - and a coincidence property for so-called von Neumann relations. The method avoids the use of deeper results of topological essence such as the Brouwer fixed point theorem or the Sperner's lemma and underlines the crucial role played by convexity. It turns out that the convex KKM principle is equivalent to the Hahn-Banach theorem, the Markov-Kakutani fixed point theorem, and the Sion-von Neumann minimax principle

Helly's Theorem: New Variations and Applications

This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.Comment: 40 pages, 1 figure

A unified theory of cone metric spaces and its applications to the fixed point theory

In this paper we develop a unified theory for cone metric spaces over a solid vector space. As an application of the new theory we present full statements of the iterated contraction principle and the Banach contraction principle in cone metric spaces over a solid vector space.Comment: 51 page

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figure

Dynamics of non-metric manifolds

An attempt is made to extend some of the basic paradigms of dynamics, from the viewpoint of (continuous) flows, to non-metric manifolds.Comment: 29 pages, 4 figure

Continuous selections of multivalued mappings

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume

Colorful theorems for strong convexity

We prove two colorful Carath\'eodory theorems for strongly convex hulls, generalizing the colorful Carat\'eodory theorem for ordinary convexity by Imre B\'ar\'any, the non-colorful Carath\'eodory theorem for strongly convex hulls by the second author, and the "very colorful theorems" by the first author and others. We also investigate if the assumption of a "generating convex set" is really needed in such results and try to give a topological criterion for one convex body to be a Minkowski summand of another

Embedding theorems for quasitoric manifolds

The study of embeddings of smooth manifolds into Euclidean and projective spaces has been for a long time an important area in topology. In this paper we obtain improvements of classical results on embeddings of smooth manifolds, focusing on the case of quasitoric manifolds. We give explicit constructions of equivariant embeddings of a quasitoric manifold described by combinatorial data $(P,\Lambda)$ into Euclidean and complex projective space. This construction provides effective bounds on the dimension of the equivariant embedding

Average-Value Tverberg Partitions via Finite Fourier Analysis

The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an $N(q,d):=(q-1)(d+1)$-simplex to $d$-dimensional Euclidian space, the existence of $q$ pairwise disjoint subfaces whose images have non-empty $q$-fold intersection. The affine cases, true for all $q$, constitute Tverberg's famous 1966 generalization of the classical Radon's Theorem. Although established for all prime powers in 1987 by \"Ozaydin, counterexamples to the conjecture, relying on 2014 work of Mabillard and Wagner, were first shown to exist for all non-prime-powers in 2015 by Frick. Starting with a reformulation of the topological Tverberg conjecture in terms of harmonic analysis on finite groups, we show that despite the failure of the conjecture, continuous maps \textit{below} the tight dimension $N(q,d)$ are nonetheless guaranteed $q$ pairwise disjoint subfaces -- including when $q$ is not a prime power -- which satisfy a variety of "average value" coincidences, the latter obtained as the vanishing of prescribed Fourier transforms.Comment: 9 pages; to appear in Israel J. Math. Final version eliminates some typo