Topology of the hyperspace of convex bodies of constant width

The hyperspace of all convex bodies of constant width in Euclidean spaceR ⁿ ,n≥2, is proved to be homeomorphic to a contractibleQ-manifold (Q denotes the Hilbert cube). The proof makes use of an explicitly constructed retraction of the entire hyperspace of convex bodies on the hyperspace of convex bodies of constant width. Translated fromMaternaticheskie Zametki, Vol. 62, No. 6, pp. 813–819, December, 1997 Translated by V. N. Dubrovsky     

On topological properties of the Hartman-Mycielski functor

We investigate some topological properties of a normal functorH introduced earlier by Radul which is some functorial compactification of the Hartman-Mycielski construction HM. We prove that the pair (H X, HMY) is homeomorphic to the pair (Q, σ) for each nondegenerated metrizable compactumX and each denseσ-compact subsetY.     

Superextensions which are Hilbert cubes

It is shown that each separable metric, not totally disconnected, topological space admits a superextension homeomorphic to the Hilbert cube. Moreover, for simple spaces, such as the closed unit interval or then-spheresS _n , we give easily described subbases for which the corresponding superextension is homeomorphic to the Hilbert cube.     

A Hilbert cube compactification of the function space with the compact-open topology

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)^ℕ of the Hilbert cube Q = [−1, 1]^ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $ \bar C $ \bar C (X) of C(X) such that the pair ($ \bar C $ \bar C (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $ \bar C $ \bar C (X) coincides with the space USCC _F (X,      

Finite dimensional polish spaces are extreme boundaries of convex bodies in Euclidean space

It is proved that everyn-dimensional Polish space is homeomorphic to the set of extreme points of a compact convex set inR ^18(n+1). The contribution of M. Levin to this paper is a part of his Ph.D. thesis prepared at the University of Haifa under the supervision of Y. Sternfeld.     

A recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes

For a given pair of finite point setsP andQ in some Euclidean space we consider two problems: Problem 1 of finding the minimum Euclidean norm point in the convex hull ofP and Problem 2 of finding a minimum Euclidean distance pair of points in the convex hulls ofP andQ. We propose a finite recursive algorithm for these problems. The algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations.     

The AR-property for Roberts' example of a compact convex set with no extreme points Part 1: general result

We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR.

     

On the topological complexity of DC-sets

A DC-set is a set defined by means of convex constraints and one additional inverse convex constraint. Given an arbitrary closed subsetM of the Euclideann-space, we show that there exists a DC-set in (n+1)-space being homeomorphic to the given setM. Secondly, for any fixedn2, we construct a compactn-dimensional manifold with boundary, which is a DC-set and which has arbitrarily large Betti-numbersr _k fork n–2.     

On the Geometry of Dual Pairs

The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space. The proof begins with a known result showing the representability of every vector as a sum of two orthogonal vectors, one coming from a cone and the other from its dual (a generalization of representation by orthogonal subspaces). The key theorem, showing that every non-zero vector has a positive multiple which is the sum of two v-dual vectors, follows from this and in turn provides the required homeomorphism. One consequence of this topological equivalence is the arc-connectedness of the numerical range determined by v.     

On products with the unit interval

We investigate the question which compact convex sets are homeomorphic to their product with the unit interval. We prove it in particular for the space of probability measures on any infinite scattered compact space and for the half-ball of a non-separable Hilbert space equipped with the weak topology. We also show examples of compact spaces for which it is not the case.     

A multidimensional real Schwarz Lemma for the Hilbert metric

We present a real multidimensional version of the Schwarz Lemma on a bounded convex domain D of ℝ ⁿ endowed with the Hilbert metric. We provide as an application an extension of a Birkhoff’s Theorem on mappings contracting the Hilbert metric.     

New topological models for Banach-Mazur compacta

Sufficient conditions are given for an action of the orthogonal group O(n) on the Hilbert cube Q in order that the corresponding orbit space Q/O(n) be homeomorphic to the Banach-Mazur compactum BM(n). This result is applied to obtain simple topological models for BM(2). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 19–31, 2005.     

Inspheres and inner products

Among normed linear spacesX of dimension ≧3, finite-dimensional Hilbert spaces are characterized by the condition that for each convex bodyC inX and each ballB of maximum radius contained inC,B’s center is a convex combination of points ofB ∩ (boundary ofC). Among reflexive Banach spaces of dimension ≧3, general Hilbert spaces are characterized by a related but weaker condition on inscribed balls. Research of the first author was partially supported by the U.S. National Science Foundation. Research of the second and third authors was supported by the Consiglio Nazionale delle Ricerche and the Ministero della Pubblica Istruzione of Italy, while they were visiting the University of Washington, Seattle, USA.     

Linear homeomorphisms of non-classical Hilbert spaces

Let K be a complete infinite rank valued field. In [4] we studied Norm Hilbert Spaces (NHS) over K i.e. K-Banach spaces for which closed subspaces admit projections of norm ≤ 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms are bijective, no NHS is linearly homeomorphic to a proper subspace (Theorem 3.7), each operator can be approximated, uniformly on bounded sets, by finite rank operators (Theorem 3.8). These properties together — in real or complex theory shared only by finite-dimensional spaces — show that NHS are more ‘rigid’ than classical Hilbert spaces.     

On unitary representability of topological groups

We prove that the additive group (E*, τ _k (E)) of an -Banach space E, with the topology τ _k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ _k (E)) is uniformly homeomorphic to a subset of for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces. Research partially supported by Spanish Ministry of Science, grant MTM2008-04599/MTM. The foundations of this paper were laid during the author’s stay at the University of Ottawa supported by a Generalitat Valenciana grant CTESPP/2004/086.     

Hilbert C*-Modules over Monotone Complete C*-Algebras

The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ?? over monotone complete C*-algebras A by the completeness of the unit ball of ?? with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182 , 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ?? can be continued to an A-valued inner product on it's A-dual Banach A-module ??' turning ??' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End ′(??) on self-dual Hilbert A-modules ?? over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.     

On locally convex hypersurfaces in Hadamard manifolds

Locally convex compact hypersurfaces immersed in a hollow simply connected Riemannian space of nonpositive sectional curvature are considered. They are proved to be convex hypersurfaces homeomorphic to the sphere. A similar result for immersed hypersurfaces with nonpositive definite second quadratic form of rank no smaller than one is obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 498–507, April, 2000.     

Discrete hilbert transforms and rearrangement invariant sequence spaces

A pair of rearrangement inequalities are obtained for a discrete analogue of the Hilbert transform which lead to necessary and sufficient conditions for certain discrete analogues of the Hilbert transform to be bouonded as linear operators between rearrangement invariant sequence spaces. In particular, if X is a rearrangement invariant space with indices α and β, then 0<β≤α<1 is both necessary and sufficient for these transforms to be bounded from X into itself, which generalizes a well known result of M. Riesz. Applications are made to discerete Hilbert transforms in higher dimensions, in particular, the discrete Riesz transforms are bounded from X into itself if and only if 0<β≤α<1.     

Division and k-th root theorems for Q-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f:K→X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X×[0,1] are Q-manifolds as well. We construct a countable familyχof spaces with DDP and cd-AP such that no space X∈χis homeomorphic to the Hilbert cube Q whereas the product X×Y of any different spaces X, Y∈χis homeomorphic to Q. We also show that no uncountable familyχwith such properties exists.     

The topological space of all extreme points of a compact convex set

G. Choquet and R. Haydon have proved that every Polish (i.e. completely metrizable and separable) spaceX is homeomorphic to the space of all extreme points of a metrizable compact convex setK which can be chosen to be even a Choquet simplex. In our paper we generalize partially this result to the case of a complete metric spaceX of zero covering dimension, without requiring any separability property forX.