The extension property for compact convex sets

A closed convex subsetQ of a compact convex setK is said to have the extension property if every continuous affine function onQ can be extended to a continuous affine function onK. It is proved that the extension property is equivalent to the existence of a numberN such that is any direction in whichQ has positive width, the ratio of the width ofK to the width ofQ is less thanN.     

Actions of compact groups on coherent sheaves

We consider actions of compact real Lie GroupsK on complex spacesX such that the associated reducedK-space admits a semistable quotient, e.g.X is a Stein space. We show that there is a complex spaceX ^c endowed with a holomorphic action of the universal complexificationG ofK that containsX as an openK-stable subset. As our main result, we prove that every coherentK-sheaf onX extends uniquely to a holomorphicG-sheaf onX ^c .Supported by a Heisenberg Stipendium of the Deutsche Forschungsgemeinschaft.     

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Every continuous function on a compact, holomorphically convex, real-analytic subset of can be approximated uniformly by functions holomorphic on the set.

     

Positivity notions for coherent sheaves over compact complex spaces

LetX be a reduced compact complex space, X a coherent sheaf, andV=V() its associated linear fiber space. LetV _R be the reduction ofV, letA be the analytic set inX over which is not locally-free, and letV be the closure inV _R ofV _R |(X–A). is (primary) weakly positive if the zerosection ofV (V) is exceptional. is (primary) cohomologically positive if, for any coherent sheaf X, for all 0,k1. Then is (primary) weakly positive if and only if is (primary) cohomologically positive.LetX be a normal irreducible compact complex space. ThenX is Moishezon if and only if it carries a primary weakly positive, and hence primary cohomologically positive, coherent sheaf.Several other positivity notions are also discussed.     

Divisible points of compact convex sets

By associating with an affine dependence the resultant of a related probability measure, we are able to define the set ofdivisible points, D(K), of a compact convex setK. Some general properties ofD(K) are discussed, and its equivalence with a set recently introduced by Reay for convex polytopes demonstrated. For polytopes,D(K) is a continuous image of a projective space. A conjecture concerningD(K) is settled affirmatively for cubes.     

Hyperspace of max-plus convex compact sets

The hyperspace mpcc(? ⁿ ) of max-plus convex compact subsets of ? ⁿ , n ≥ 2, is considered. The main result is as follows: this hyperspace is a contractible manifold modeled on the Hilbert cube Q. It is also shown that the projection mapping mpcc(? ⁿ ) → mpcc(? ^m ), n ≥ m, is open. Moreover, it is proved that the hyperspace mpcc( $ I^{\omega _1 } $ ) of the Tikhonov [Tychonoff] cube $ I^{\omega _1 } $ is homeomorphic to $ I^{\omega _1 } $ .     

The facialQ-topology for compact convex sets

The structure of the set of closed two-sided ideals in aC*-algebraU with identity is described by means of a topology on the set _e K of extreme points of the state spaceK ofU. Recent results of Alfsen, Andersen, Combes, Perdrizet, Wils, and others have shown that such a topology can be defined on the set _e K of extreme points of an arbitrary compact convex subset of a locally convex Hausdorff topological vector space.The structure of the set of closed left ideals in aC*-algebraU with identity can also be described by means of a set of subsets of the set _e K of extreme points of its state spaceK. Akemann, Giles, and Kummer showed that this formed a more general structure than a topology which was called aq-topology. In this paper it is shown that for a reasonably wide class of compact convex subsetsK of locally convex Hausdorff topological vector spaces such aq-topology can also be defined on _e K and that it shares many of the properties of theq-topology defined forC*-algebras. The methods used depend strongly upon recent results of Alfsen and Shultz on the spectral theory of affine functions on compact convex sets.     

Numerical range and compact convex sets

Let K be a compact convex subset of the plane, μ be a regular Borel measure with support K and N _μ be the multiplication operator on L ²(μ). In this article we show that \(\overline{W}(N_{\mu})\), the closure of numerical range of N _μ , is K. Also we prove that if K has uncountable many extreme points then the Berberian Hilbert space extension of L ²(μ) is non separable.     

Weights of boundaries of compact convex sets

Let B be a boundary of a compact convex set X. We prove that the weight of X equals the weight of B provided B is Lindelöf or X is a standard compact convex set and B is the set of extreme points of X.     

Boundaries of compact convex sets and fragmentability

If X is a compact convex set in a real locally convex space, B⊂X is said to be its boundary if every affine continuous function on X attains its maximum at some point of B. We study relations between fragmentability of B and the whole set X. As a byproduct we obtain a characterization of separable Asplund spaces. We also study the possibility of finding the Haar system in a boundary of a metrizable compact convex set.     

Minimizing pseudoconvex functions on convex compact sets

An algorithm is presented which minimizes continuously differentiable pseudoconvex functions on convex compact sets which are characterized by their support functions. If the function can be minimized exactly on affine sets in a finite number of operations and the constraint set is a polytope, the algorithm has finite convergence. Numerical results are reported which illustrate the performance of the algorithm when applied to a specific search direction problem. The algorithm differs from existing algorithms in that it has proven convergence when applied to any convex compact set, and not just polytopal sets.This research was supported by the National Science Foundation Grant ECS-85-17362, the Air Force Office Scientific Research Grant 86-0116, the Office of Naval Research Contract N00014-86-K-0295, the California State MICRO program, and the Semiconductor Research Corporation Contract SRC-82-11-008.     

The compact Z-set property in convex sets

It is shown that in very metrizable infinite-dimensional convex set C which is a countable union of finite-dimensional compacta all compacta are Z-sets. As an application, some conditions are found in order for C to be hemeomorphic to l^f₂.