Separated totally convex spaces

In this paper we define for every totally convex space a suitable topology, the radial topology. We prove that a morphism in the category TC^sep of separated totally convex spaces is an epimorphism if and only if its image is dense in the radial topology, and that TC^sep is the full subcategory of TC generated by its Hausdorff objects. These results remain valid for finitely totally convex spaces when the radial topology is replaced by the distance-radial topology.Dedicated to Karl Stein     

What is concrete equivalence?

The notion of concrete equivalence is introduced, based on a modification of the traditional notion of concrete functor. The discussion of examples includes a direct (i.e. not referring to any monadicity theorem) proof of the fact that monadicity is stable under concrete equivalence.For the sixtieth birthday of Nico PumplünHospitality of the Department of Mathematics, Applied Mathematics and Astronomy at UNISA is gratefully acknowledged, where this note was completed during an extended visit.     

More on directed colimits of models

M. Richter has proved that whenever a classK of -structures has a finitary first-order axiomatization then the inclusionK Str preserves all existing directed colimits (see [7]). We will generalize this result to classes of -structures having an infinitary first-order axiomatization in a larger signature . We will also show that, as categories, these classes have a natural characterization.In honour of Dieter Pumplün on the occassion of his 60th birthday     

Convex bodies of constant width and constant brightness

In 1926 Nakajima (= Matsumura) showed that any convex body in R³ with constant width, constant brightness, and boundary of class C² is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.     

Constant morphisms and constant subcategories

In a category supplied with a factorization system for morphisms and a fixed subcategory of constant objects, we introduce suitable notions ofconstant morphism and of the correspondingright andleft constant subcategories. The nature of constant morphisms we use captures two important features of constant subcategories: left-constant subcategories are right-constant in the dual category and the subcategory of constant objects contains relevant information on these subcategories. Furthermore, we present characterizations of constant subcategories in several contexts. Namely, we extend the characterization of disconnectednesses obtained by Huek and Pumplün, via terminal fans, to our context.The author acknowledges financial support by Centro de Matemática da Universidade de Coimbra.     

Onm-separated projection spaces

Closure operators in the category of projection spaces are investigated. It is shown that completeness, absolutes-closure ands-injectivity coincide in the subcategory of separated projection spaces and that there compactness with respect to projections implies completeness.Dedicated to Nico Pumplün on the occasion of his 60th birthdayPartial financial support by the Italian Ministry of Public Education is gratefully acknowledged.     

Regular closure operators

In an E,M-categoryX for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms inM to factor through the lattice of all closure operators onM, and to factor through certain sublattices. This leads to the notion ofregular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün-Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class ofX-objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.Dedicated to Professor Dieter Pumplün, on his 60th birthdayResearch partially supported by the Faculty of Arts and Sciences, University of Puerto Rico, Mayagüez Campus during a sabbatical visit at Kansas State University.     

A cogenerator for preseparated superconvex spaces

We construct a cogenerator for the category of preseparated superconvex spaces, and we describe separated convex spaces, i.e. convex spaces for which the morphisms into the unit interval separates points.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Duality of convex bodies

The paper presents a category theoretical approach to the notion of duality of convex bodies. Using results of I. Barany (Acta Sci. Math. (Szeged)52 (1988), 93–100), we define and study metric duality , whose advantage is that congruent convex bodies have congruent duals.Dedicated to Professor Helmut Salzmann on the occasion of his 65th birthday     

Almost coreflective and projective classes

This paper summarizes the situation around the problem of when classes of projective objects are almost coreflective, both in general categories and in Top or similar categories. In addition to known results, several new contributions and examples are added.For the sixtieth birthday of D. PumplünThe paper was written while the second author was visiting the University of Toledo, Ohio.     

Convex bodies with extremal volumes having prescribed brightness in finitely many directions

We consider the class of convex bodies in ⁿ with prescribed projection (n – 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximumn-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions.Conditions for the existence of bodies with minimumn-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established.Work partially supported by Istituto di Analisi Globale e Applicazioni, CNR, Firenze.     

On closed convex sets without boundary rays and asymptotes

We study in finite-dimensional spaces the class of closed convex sets without boundary rays and asymptotes, denoted by and introduced by D. Gale and V. Klee. These sets, not necessarily bounded, enjoy many properties satisfied by compacts sets. New properties of this class are given and convergence analysis of this class is investigated. We also introduce the class of closed convex proper functions which have an epigraph in and we give some properties of these functions.     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

How accessible are categories of algebras?

For locally finitely presentable categories it is well known that categories of F-algebras, where F is a finitary endofunctor, are also locally finitely presentable. We prove that this generalizes to locally finitely multipresentable categories. But it fails, in general, for finitely accessible categories: we even present an example of a strongly finitary functor F (one that preserves finitely presentable objects) whose category of F-algebras is not finitely accessible. On the other hand, categories of F-algebras are proved to be ω₁-accessible for all strongly finitary functors—and it is an open problem whether this holds for all finitary functors.     

New affine measures of symmetry for convex bodies

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are “far apart”. In particular, we give an example of a convex body whose centroid and Santaló point are “far apart”.     

Tilings of polygons with similar triangles

We prove that if a polygonP is decomposed into finitely many similar triangles then the tangents of the angles of these triangles are algebraic over the field generated by the coordinates of the vertices ofP. IfP is a rectangle then, apart from four sporadic cases, the triangles of the decomposition must be right triangles. Three of these sporadic triangles tile the square. In any other decomposition of the square into similar triangles, the decomposition consists of right triangles with an acute angle such that tan is a totally positive algebraic number. Most of the proofs are based on the following general theorem: if a convex polygonP is decomposed into finitely many triangles (not necessarily similar) then the coordinate system can be chosen in such a way that the coordinates of the vertices ofP belong to the field generated by the cotangents of the angles of the triangles in the decomposition.This work was completed while the author had a visiting position at the Mathematical Institute of the Hungarian Academy of Sciences.     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

Banach's Fixed-Point Theorem as a base for data-type equations

For a categoryK of data types, solutions of recursive data-type equationsX T(X), whereT is an endofunctor ofK, can be constructed by iteratingT on the unique arrowT1 1. This is well-known forK enriched over complete posets and forT locally continuous (an application of the Kleene Fixed-Point Theorem). We prove this forK enriched over complete metric spaces and forT contracting (an application of the Banach Fixed-Point Theorem). Moreover, we prove that each such recursive equation has a unique solution. Our results generalize the approach of P. America and J. Rutten.Dedicated to Dieter Pumplün on the occasion of his 60th birthday