Continuity properties of the ball hull mapping

The ball hull mapping  β$\beta$ associates with each closed bounded convex set K$K$ in a Banach space its ball hull β(K)$\beta \left(K\right)$, defined as the intersection of all closed balls containing K$K$. We are concerned in this paper with continuity and Lipschitz continuity (with respect to the Hausdorff metric) of the ball hull mapping. It is proved that β$\beta$ is a Lipschitz map in finite dimensional polyhedral spaces. Both properties, finite dimension and polyhedral norm, are necessary for this result. Characterizing the ball hull mapping by means ofH$H$-convexity we show, with the help of a remarkable example from combinatorial geometry, that there exist norms with noncontinuous β$\beta$ map, even in finite dimensional spaces. Using this surprising result, we then show that there are infinite dimensional polyhedral spaces (in the usual sense of Klee) for which the map β$\beta$ is not continuous. A property known as ball stability implies that β$\beta$ has Lipschitz constant one. We prove that every Banach space of dimension greater than two can be renormed so that there is an intersection of closed balls for which none of its parallel bodies is an intersection of closed balls, thus lacking ball stability.     

Convex bodies with equiframed two-dimensional sections

We show that a centred, convex body in (d ≥ 3) all of whose two-dimensional sections through the origin are equiframed is an ellipsoid. Received: 20 July 2005 Revised: 16 May 2006     

A problem of Kusner on equilateral sets

R. B. Kusner [R. Guy, Amer. Math. Monthly 90, 196-199 (1983)] asked whether a set of vectors in such that the distance between any pair is 1, has cardinality at most d + 1. We show that this is true for p = 4 and any , and false for all 1<p<2 with d sufficiently large, depending on p. More generally we show that the maximum cardinality is at most if p is an even integer, and at least if 1<p<2, where depends on p. Received: 5 May 2003     

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.     

On the nontrivial projection problem

The nontrivial projection problem asks whether every finite-dimensional normed space admits a well-bounded projection of nontrivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension) is approximately affinely equivalent to a direct product of two bodies of nontrivial dimensions. We show that this is true “up to a logarithmic factor.”     

Quantitative Steiner/Schwarz-type symmetrizations

We establish some new quantitative results on Steiner/Schwarz-type symmetrizations, continuing the line of results from [Bourgain et al. (Lecture Notes in Math. 1376 (1988), 44–66)] on Steiner symmetrizations. We show that if we symmetrize high-dimensional sections of convex bodies, then very few steps are required to bring such a body close to a Euclidean ball.     

Belousov equations on ternary quasigroups

Summary The study of Belousov equations in binary quasigroups was initiated by V. D. Belousov. Krape and Taylor showed that every finite set of Belousov equations was equivalent to a single Belousov equation which was in some sense no longer than any single member of the set. This led to the concept of an irreducible Belousov equation, that is one which is not equivalent to an equation with fewer variables. Krape and Taylor determined the structure of the irreducible equations by establishing a correspondence between them and specific polynomials overZ ₂.In this paper it is shown that the structure of the ternary equations is richer than the binary counterpart, although the main result is similar to the binary case in as far as a system of ternary Belousov equations is equivalent to a single Belousov equation which is no longer than any member of the system or the system is equivalent to a pair of equations each with three variables.     

Extremal structure of the unit ball of direct sums of Banach spaces

We characterize the extreme points and smooth points of the unit ball of certain direct sums of Banach spaces. We use these results to characterize noncreasiness and uniform noncreasiness of direct sums, thereby extending results of the second author [S. Saejung, Extreme points, smooth points and noncreasiness of ψ$\psi$-direct sum of Banach spaces, Nonlinear Anal. 67 (2007) 1649–1653].     

Swimming below icebergs

You are swimming close to an iceberg with a convex lower surface. You calculate at what slope you have to swim down so that, whatever the direction in which you swim, you can be sure that you will not collide with the iceberg. This limiting slope is intimately related to the existence of subtangents to the iceberg that satisfy various conditions. These considerations lead to generalizations of Rockafellar's Maximal Monotonicity Theorem, of which we give acomplete new proof. We also discuss related open problems on maximal monotonicity and subdifferentials, and generalizations of recent results on the existence of subtangents separating the epigraphs of proper convex lower semicontinuous functions from nonempty bounded closed convex sets, with some control over their slopes.     

Application of an idea of Vorono? to John type problems

Using an idea of Vorono?, many John type and minimum position problems in dimension d can be transformed into more accessible geometric problems on convex subsets of the -dimensional cone of positive definite quadratic forms. In this way, we prove several new John type and minimum position results and give alternative versions and extensions of known results. In particular, we characterize minimum ellipsoidal shells of convex bodies and, in the typical case, show their uniqueness and determine the contact number. These results are formulated also in terms of the circumradius of convex bodies. Next, circumscribed ellipsoids of minimum surface area of a convex body and the corresponding minimum position problem are studied. Then we investigate John type characterizations of minimum positions of a convex body with respect to moments and the product of a moment and the moment of the polar body. The technique used in this context, finally, is applied to obtain corresponding results for the mean width and the surface area.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

Wijsman convergence: A survey

A net A of nonempty closed sets in a metric space X, d is declaredWijsman convergent to a nonempty closed setA provided for eachx X, we haved(x, A)=lim d(x, A). Interest in this convergence notion originates from the seminal work of R. Wijsman, who showed in finite dimensions that the conjugate map for proper lower semicontinuous convex functions preserves convergence in this sense, where functions are identified with their epigraphs. In this paper, we review the attempts over the last 25 years to produce infinite-dimensional extensions of Wijsman's theorem, and we look closely at the topology of Wijsman convergence in an arbitrary metric space as well. Special emphasis is given to the developments of the past five years, and several new limiting counterexamples are presented.     

Every Banach Space is Reflexive

The title above is wrong, because the strong dual of a Banach space is too strong to assert that the natural correspondence between a space and its bidual is an isomorphism. However, for many applications it suffices to replace the norm on the first dual by the weak*-structure in order to solve the non-reflexiveness problem [1]. But in this way, only the original vector space is recovered by taking the second dual. In this work we introduce a suitable numerical structure on vector spaces such that Banach balls, or more precisely totally convex modules, arise naturally in duality, namely as a category of Eilenberg–Moore algebras. This numerical structure naturally overlies the weak*-topology on the algebraic dual, so the entire Banach space can be reconstructed as a second dual. Moreover, the isomorphism between the original space and its bidual is the unit of an adjunction between the two-dualisation functors. Notice that the weak*-topology is normable only if it lives on a finite dimensional space; in that case the original space is trivial as well, hence reflexive. So the overlying numerical structure should be something more general than a norm or a seminorm and thus approach theory [2, 3] enters the picture.