Matching binary convexities

We showed in an earlier paper that the Radon number of an n-dimensional binary convexity equals the Radon number of the n-cube, except for a well-determined sequence of dimensions, in which case the Radon number may be one unit larger. Examples of the latter are obtained in every predicted dimension. The basic tool is a matching procedure which works for binary convexities.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.     

The diagonal hypergraph of a matrix (bipartite graph)

Let A be an n x n matrix of 0's and 1's (a bipartite graph). The diagonal hypergraph of A is the hypergraph whose vertices correspond to the 1's (edges) of A and whose edges correspond to the positive diagonals (1-factors) of A. The numerical invariants of this hypergraph are investigated.     

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ<Δ(X), where Δ(X) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.     

Fractal boundaries are not typical

Let M be a C¹n-dimensional compact submanifold of Rn. The boundary of M, ∂M, is itself a C¹ compact (n−1)-dimensional submanifold of Rn. A carefully chosen set of deformations of ∂M defines a complete subspace consisting of boundaries of compact n-dimensional submanifolds of Rn, thus the Baire Category Theorem applies to the subspace. For the typical boundary element ∂W in this space, it is the case that ∂W is simultaneously nowhere-differentiable and of Hausdorff dimension n−1.     

Capacities, surface area, and radial sums

A dual capacitary Brunn-Minkowski inequality is established for the (n−1)-capacity of radial sums of star bodies in Rn. This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the p-capacity of Minkowski sums of convex bodies in Rn, 1?p<n, proved by Borell, Colesanti, and Salani. When n?3, the dual capacitary Brunn-Minkowski inequality follows from an inequality of Bandle and Marcus, but here a new proof is given that provides an equality condition. Note that when n=3, the (n−1)-capacity is the classical electrostatic capacity. A proof is also given of both the inequality and a (different) equality condition when n=2. The latter case requires completely different techniques and an understanding of the behavior of surface area (perimeter) under the operation of radial sum. These results can be viewed as showing that in a sense (n−1)-capacity has the same status as volume in that it plays the role of its own dual set function in the Brunn-Minkowski and dual Brunn-Minkowski theories.     

Detecting codimension one manifold factors with 0-stitched disks

The 0-stitched disks property is introduced and shown to detect codimension one manifold factors of dimension n?4. It is shown that if a space X is an ANR and has the 0-stitched disks property, then X has the disjoint homotopies property. It follows that if a space X is a resolvable generalized manifold of dimension n?4 with the 0-stitched disks property, then X is a codimension one manifold factor. Whether or not the 0-stitched disks property is equivalent to the disjoint homotopies property remains an open question.     

Sacks forcing and the total failure of Martin's Axiom

Using side-by-side Sacks forcing, it is proved relatively consistent that the continuum is large and Martin's Axiom fails totally, that is, every c.c.c. space is the union of ?₁ nowhere dense sets (equivelently, if P is a nontrivial partial ordering with the countable chain condition, then there are ?₁ dense sets in P such that no filter in P meets them all).     

Equiframed Curves – A Generalization of Radon Curves

Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their higher-dimensional analogues were introduced by Peczynski and Szarek (1991, Math Proc Cambridge Philos Soc 109: 125–148). Radon curves form a proper subclass of this class of curves. Our main result is a construction of an arbitrary equiframed curve by appropriately modifying a Radon curve. We give characterizations of each type of curve to highlight the subtle difference between equiframed and Radon curves and show that, in some sense, equiframed curves behave dually to Radon curves.Research supported by a grant from a cooperation between the Deutsche Forschungsgemeinschaft in Germany and the National Research Foundation in South Africa. Parts of this paper were written during a visit to the Department of Mathematics, Applied Mathematics and Astronomy of the University of South Africa.     

Covering dimension and finite-to-one maps

Hurewicz characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most n²-to-1 map.     

Geometric topological completions with universal final lifts

In this paper necessary and sufficient conditions are given on a concrete category over a category B so that it can be densely embedded (over B) into a geometric topological category E that admits certain universal final lifts. These conditions, as well as the class of universal final lifts, depend upon an a priori given full subcategory Δ of B. For example, E may have, depending upon Δ and B, universal coproducts or quotients or colimits. For appropriate Δ's, if B is cartesian closed then so is E.     

Extension theorems for some classes of continua

It is proved that every mapping from a proper subcontinuum of a hereditarily unicoherent continuum X onto the Knaster's indecomposable continuum (onto a cone over a zerodimensional compact metric set) can be extended to a mapping defined on X.Similarly, every mapping from a proper subcontinuum of a hereditarily indecomposable continuum onto a pseudoarc can be extended to a mapping defined on the whole space.Both of the above results are generalizations of the author's earlier results to the nonmetric case. As a consequence it is obtained that a pseudoarc is continuously n-homogeneous.     

Iterations of linear maps over finite fields

We study the dynamics of the evolution of Ducci sequences and the Martin-Odlyzko-Wolfram cellular automaton by iterating their respective linear maps on . After a review of an algebraic characterization of cycle lengths, we deduce the relationship between the maximal cycle lengths of these two maps from a simple connection between them. For n odd, we establish a conjugacy relationship that provides a more direct identification of their dynamics. We give an alternate, geometric proof of the maximal cycle length relationship, based on this conjugacy and a symmetry property. We show that the cyclic dynamics of both maps in dimension 2n can be deduced from their periodic behavior in dimension n. This link is generalized to a larger class of maps. With restrictions shared by both maps, we obtain a formula for the number of vectors in dimension 2n belonging to a cycle of length q that expresses this number in terms of the analogous values in dimension n.     

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.     

Some properties of C(X), I

By a result of A.V. Arhangel'skiǐ and E.G. Pytkeiev, the space C(X) of the continuous real functions on X with the topology of pointwise convergence has tightness ω iff Xⁿ is Lindelöf for every n ∈ ω. In this paper we describe other convergence properties of C(X) (e.g. the Fréchet-Urysohn properly) in terms of covering properties of X.In some cases the equivalence between these properties turn out to be dependent on the set theory we choose. Some open problems are also stated.     

Equidissections of centrally symmetric octagons

Summary In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1988 Kasimatis proved that if a regularn-gon,n 5, is cut intom triangles of equal areas, thenm is a multiple ofn. These two results imply that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas. We conjecture that the conclusion holds even if the restriction regular is deleted from the hypothesis and prove that it does forn = 6 andn = 8.     

Turán theorems and convexity invariants for directed graphs

This paper is motivated by the desire to evaluate certain classical convexity invariants (specifically, the Helly and Radon numbers) in the context of transitive closure of arcs in the complete digraph. To do so, it is necessary to establish several new Turán type results for digraphs and characterize the associated extremal digraphs. In the case of the Radon number, we establish the following analogue for transitive closure in digraphs of Radon's classical convexity theorem [J. Radon, Mengen konvexer Körper, die einer gemeinsamen Punkt enthalten, Math. Ann. 83 (1921) 113-115]: in a complete digraph on n?7 vertices with >n²/4 arcs, it is possible to partition the arc set into two subsets whose transitive closures have an arc in common.     

Intersection bodies and generalized cosine transforms

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of λ-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto lower-dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms, and sections of λ-intersection bodies is established. New integral formulas for different classes of cosine transforms are obtained and examples of λ-intersection bodies are given. We also revisit some known facts in this area and give them new simple proofs.     

The notion of o-tightness and C-embedded subspaces of products

We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Π_α?AX_α has a countable o-tightness and Y is a subset of X such that π_B(Y) = Π_α?BX_α for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding.