Intersections by hyperplanes

LetA be a finite nonempty family of nonempty disjoint closed and bounded sets in a Banach spaceE which is either separable and the conjugate of some Banach spaceX (i.e.E=X*) or, reflexive and locally uniformly convex. IfC denotes the weak*-closed convex hull of ∪ {A:A ∈A} then the set of points inE ∼C through which there is no hyperplane intersecting exactly one member ofA is discrete (or empty). This research was supported by the National Research Council of Canada, Grant A-3999.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

Intersectional properties concerning planarm-convex sets

LetC be a collection of closed sets in the plane, and letS=∩C. (1) If IncC ⊆ kerS for allC inC and if dim kerS≧1, thenS is a union of three (or fever) convex sets. In particular, the results holds when the members ofC are 3-convex sets, all having the same kernelK, provided dimK≧1. (2) IfC is a finite collection ofm-convex sets such that ∩{kerC:C inC inC} ≠ ⊘,S~ IncS is connected, and for someZ inC, lncC⊆ lncZ for allC inC, thenS ism-convex.     

Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰ_ub(E), of allC ^∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰ_Nbc(E _v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ^∞-functions onE coincides with the projective limit of the spaces ℰ_Nbc(E _B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem. Research done during the stay of this author at the University of Bordeaux (France) in the academic year 1980–1981.     

Non-itpfidiffeomorphisms

We constructC ^x diffeomorphisms ofT ⁴ which give rise via the group measure space construction to factors which are not ITPFI. We extend the construction to arbitrary paracompact, connected manifolds of dimension ≧6. Research partially supported by NSF Grant #MCS 8102399.     

Projections of 3-polytopes

Given any 3-dimensional convex polytopeP, and any simple circuitC in the 1-skeleton ofP, there is a convex polytopeP′ combinatorially equivalent toP, and a direction such that ifP′ is projected orthogonally in this direction, then the inverse image of the boundary of the projection is the circuit inP′ corresponding to the circuitC inP. Research supported by NSF Grant GP-8470.     

Guarding galleries where every point sees a large area

We prove a conjecture of Kavraki, Latombe, Motwani and Raghavan that ifX is a compact simply connected set in the plane of Lebesgue measure 1, such that any pointx∈X sees a part ofX of measure at least ɛ, then one can choose a setG of at mostconst1/ɛ log 1/ɛ points inX such that any point ofX is seen by some point ofG. More generally, if for anyk points inX there is a point seeing at least 3 of them, then all points ofX can be seen from at mostO(k ³ logk) points. Research supported by grants from the Sloan Foundation, the Israeli Academy of Sciences and Humanities, and by G.I.F. Research supported by Czech Republic Grant GAČR 201/94/2167 and Charles University grants No. 351 and 361. Part of the work was done while the author was visiting The Hebrew University of Jerusalem.     

First Borel class sets in Banach spaces and the asymptotic-norming property

The Radon-Nikodym property in a separable Banach spaceX is related to the representation ofX as a weak* first Borel class subset of some dual Banach space (its bidualX**, for instance) by well known results due to Edgar and Wheeler [8], and Ghoussoub and Maurey [9, 10, 11]. The generalizations of those results depend on a new notion of Borel set of the first class “generated by convex sets” which is more suitable to deal with non-separable Banach spaces. The asymptotic-norming property, introduced by James and Ho [13], and the approximation by differences of convex continuous functions are also studied in this context. Research partially supported by the grant DGES PB 98-0381.     

On compactification of metric spaces

Iff:X →X* is a homeomorphism of a metric separable spaceX into a compact metric spaceX* such thatf(X)=X*, then the pair (f,X*) is called a metric compactification ofX. An absoluteG _δ-space (F _σ-space)X is said to be of the first kind, if there exists a metric compactification (f,X*) ofX such that , whereG _i are sets open inX* and dim[Fr(G _i)]<dimX. (Fr(G _i) being the boundary ofG _i and dimX — the dimension ofX). An absoluteG _δ-space (F _σ-space), which is not of the first kind, is said to be of the second kind. In the present paper spaces which are both absoluteG _δ andF _σ-spaces of the second kind are constructed for any positive finite dimension, a problem related to one of A. Lelek in [11] is solved, and a sufficient condition onX is given under which dim [X* −f(X)]≧k, for any metric compactification (f,X*) ofX, wherek≦dimX is a given number. This research has been sponsored by the U.S. Navy through the Office of Naval Research under contract No. 62558-3315.     

Moduli of non-dentability and the radon-nikodým property in banach spaces

We show that for a separable Banach spaceX failing the Radon-Nikodym property (RNP), andε > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, forε > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −ε. Some more related results about the geometry of Banach spaces failing RNP are given.     

Low dimensional sections versus projections of convex bodies

The structure of low dimensional sections and projections of symmetric convex bodies is studied. For a symmetric convex bodyB ⊂ ℝ ⁿ , inequalities between the smallest diameter of rank ℓ projections ofB and the largest in-radius ofm-dimensional sections ofB are established, for a wide range of sub-proportional dimensions. As an application it is shown that every bodyB in (isomorphic) ℓ-position admits a well-bounded (√n, 1)-mixing operator. Research of this author was partially supported by KBN Grant no. 1 P03A 015 27. This author holds the Canada Research Chair in Geometric Analysis.     

On the topology induced by the adjoint of a semigroup of operators

The adjoint of aC ₀-semigroup on a Banach spaceX induces a locally convex σ(X,X ^ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X ^ℴ)-topology and it is shown that the uniform limit of σ(X,X ^ℴ)-compact operators is σ(X,X ^ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.     

C-semigroups and strongly continuous semigroups

We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| _Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖_[Im(C)]≡‖C ⁻¹ x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| _X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.     

A criterion for the affine equivalence of cell complexes inR d and convex polyhedra inR d+1

A criterion is given that decides, for a convex tilingC ofR ^d , whetherC is the projection of the faces in the boundary of some convex polyhedronP inR ^d+1. Its applicability is restricted neither by the properties nor by the dimension ofC. It turns out to be conceptually simpler than other criteria and allows the easy examination of various classes of cell complexes. In addition, the criterion is constructive, that is, it can be used to constructP provided it exists.Research was supported by the Austrian Fonds zur Foerderung der wissenschaftlichen Forschung.     

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

LetX(-ϱB ^m ×C ⁿ be a compact set over the unit sphere ϱB ^m such that for eachz∈ϱB ^m the fiberX _z ={ω∈C ⁿ ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC ⁿ . The polynomial hull ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z ₀,w ₀)∈Int there exists a smooth up to the boundary analytic discF:Δ→B ^m ×C ⁿ with the boundary inX such thatF(0)=(z ₀,w ₀). This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.     

The minimal cardinality where the Reznichenko property fails

A topological spaceX has the Fréchet-Urysohn property if for each subsetA ofX and each elementx inĀ, there exists a countable sequence of elements ofA which converges tox. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect each neighborhood ofx. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a setX of real numbers such thatC _p(X) does not have the weak Fréchet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space^Nℕ.) We prove the Kočinac-Scheepers conjecture by showing that ifC _p(X) has the Reznichenko property, then a continuous image ofX cannot be a subbase for a non-feeble filter on ℕ. The author is partially supported by the Golda Meir Fund and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Interpolation in the unit ball ofC n

A necessary and sufficient condition is given for a discrete multiplicity variety in the unit ballB _n ofC ⁿ to be an interpolating variety for weighted spaces of holomorphic functions inB _n . Partially supported by NSF Grant DMS-9706376.     

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X). BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332.     

Optimization of convex functions on w*-compact sets

We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw ^*-lower semicontinuous function ϕ defined on aw ^*-compact convex setC in a dual Banach spaceX ^* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw ^*−H _σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets. Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.