A Helly theorem in weakly modular space

The d-convex sets in a metric space are those subsets which include the metric interval between any two of its elements. Weak modularity is a certain interval property for triples of points. The d-convexity of a discrete weakly modular space X coincides with the geodesic convexity of the graph formed by the two-point intervals in X. The Helly number of such a space X turns out to be the same as the clique number of the associated graph. This result thus entails a Helly theorem for quasi-median graphs, pseudo-modular graphs, and bridged graphs.     

A Tverberg-type generalization of the Helly number of a convexity space

In 1966 H. Tverberg gave a far reaching generalization of the well-known classical theorem of J. Radon. In this paper a similar generalization of the classical Helly theorem is given and it is shown that among these two generalized theorems a relationship holds similar to a theorem proved by F.W. Levi in 1951. Also the generalized Helly theorem in the convex product and convex sum space are investigated.     

A Helly theorem for convexity in graphs

It is shown that for chordless path convexity in any graph, the Helly number equals the size of a maximum clique.     

A generalization of the Curtiss theorem for moment generating functions

The Curtiss theorem deals with the relation between the weak convergence of probability measures on the line and the convergence of theirmoment generating functions in a neighborhood of zero. We present a multidimensional generalization of this result. To this end, we consider arbitrary σ-finite measures whose moment generating functions exist in a domain of multidimensional Euclidean space not necessarily containing zero. We also prove the corresponding converse statement.     

On a theorem of Helly

We consider a group of problems related to the well-known Helly theorem on the intersections of convex bodies. We introduce convex subsetsK(?) of a compact convex setK defined by the relation $$K(f) = co\left\{ {\frac{N}{{N + 1}}x + \frac{N}{{N + 1}}f(x)} \right\}{\text{ }}(x \in K \subset \mathbb{R}^N ),$$ where?: K→K are continuous mappings, and prove that the intersection ∩ _?∈F K(?) is not empty; hereF is the set of all continuous mappings?: K→K.     

A fractional Helly theorem for convex lattice sets

A set of the form , where is convex and denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2^d. We prove that the fractional Helly number is only d+1: For every d and every α∈(0,1] there exists β>0 such that whenever F₁,…,F_n are convex lattice sets in such that for at least index sets I⊆{1,2,…,n} of size d+1, then there exists a (lattice) point common to at least βn of the F_i. This implies a (p,d+1)-theorem for every p?d+1; that is, if is a finite family of convex lattice sets in such that among every p sets of , some d+1 intersect, then has a transversal of size bounded by a function of d and p.     

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X^* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)^* and B(Σ, X)^** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X^*) denote the Banach space of all countably additive vector measures ν: Σ → X^* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X^*), B(Σ, X))-sequential compactness in bvca(Σ, X^*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)^*, B(Σ, X))-sequentially compact subset of B(Σ, X)_c^~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)^*, B(Σ, X)^**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)^*, B(Σ, X))-convergent sequences in B(Σ, X)_c^~ are σ(B(Σ, X)^*, B(Σ, X)^**)-convergent.     

The Berry-Esseen theorem for functions of uniform spacings

Summary A Berry-Esseen bound of order is established for suitably normalized sums of nonlinear measurable functions of uniform spacings under the natural moment assumptions.Some of this study was carried out at the Mathematical Centre, Amsterdam, and some of it when the second author was a member of the Mathematical Sciences Research Institute, Berkeley     

A note on the colorful fractional Helly theorem

Helly’s theorem is a classical result concerning the intersection patterns of convex sets in ${\mathbb{R}}^d$. Two important generalizations are the colorful version and the fractional version. Recently, Bárány et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.     

Integrals for functions with values in a partially ordered vector space

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.     

The Banach-Zarecki theorem for functions with values in metric spaces

Using an old result of Luzin about his property , we prove a general version of the Banach-Zarecki theorem (on absolute continuity and Luzin's property ).

     

A Helly theorem for geodesic convexity in strongly dismantlable graphs

A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x₀,…, x so that, for each ordinal β < , there exists a strictly increasing finite sequence (i_j)_{0 j n} of ordinals such that i₀ = β, i_n = and x_ij+1 is adjacent with x_ij and with all neighbors of x_ij in the subgraph ofG induced by {x_y: β γ }. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets.     

A Helly theorem for intersections of orthogonally starshaped sets

Let $\cal{F}$ be a finite family of simply connected orthogonal polygons in the plane. If every three (not necessarily distinct) members of $\cal{F}$ have a nonempty intersection which is starshaped via staircase paths, then the intersection $\cap \{F : F\; \hbox{in}\; \cal{F}\}$ is a (nonempty) simply connected orthogonal polygon which is starshaped via staircase paths. Moreover, the number three is best possible, even with the additional requirement that the intersection in question be nonempty. The result fails without the simple connectedness condition.     

An extension of Edelstein's theorem to a uniform space

Summary The paper contains fixed point theorems of Edelstein's type for contractive mappings in (,)-chainable uniform spaces.     

Fixed-point theorem in classes of function with values in a dq-metric space

We prove a fixed point result for nonlinear operators, acting on some classes of functions with values in a dq-metric space, and show some applications of it. The result has been motivated by some issues arising in Ulam stability. We use a restricted form of a contraction condition.