The -equation and the Hartogs phenomenon on weakly q-pseudoconcave C R manifolds

We show that the Hartogs phenomenon holds in minimal, weakly 2-pseudoconcave generic C R submanifolds of a Stein manifold with trivial normal bundle. We also prove some results concerning the local and/or global solvability of the tangential Cauchy-Riemann equations for smooth forms and currents on weakly q-pseudoconcave C R manifolds.     

Problèmes d'extension dans les domaines convexes de type fini

Résumé Soient donnés D un domaine borné de , convexe, de type fini, X une hypersurface complexe telle que soit non vide, connexe et transverse et . Nous nous intéressons au problème suivant. Sous quelle(s) condition(s) existe-t-il telle que Φ|X = φ. Nous allons donner une condition nécessaire à l'existence de l'extension Φ puis sous une condition un peu plus forte nous montrerons la continuité d'un opérateur d'extension de type Berndtsson-Andersson.      

Solving the Gleason problem on linearly convex domains

Let be a bounded, connected linearly convex set in with boundary. We show that the maximal ideal (both in ) and ) consisting of all functions vanishing at is generated by the coordinate functions . Received: 2 July 2001; in final form: 26 September 2001 / Published online: 28 February 2002     

Comparing the Bergman and Szegö projections on domains with subelliptic boundary Laplacian

We prove that the difference between the Bergman and Szegö projections on a bounded, pseudoconvex domain (with C ^∞ boundary) is smoothing whenever the boundary Laplacian is subelliptic. An equivalent statement is that the Bergman projection can be represented as a composition of the Szegö and harmonic Bergman projections (along with the restriction and Poisson extension operators) modulo an error that is smoothing. We give several applications to the study of optimal mapping properties for these projections and their difference.     

Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper, we will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.     

The Bremermann–Dirichlet problem for unbounded domains of $${\mathbb{C}}^n$$

Given an unbounded strongly pseudoconvex domain Ω and a continuous real valued function h defined on bΩ, we study the existence of a (maximal) plurisubharmonic function Φ on Ω such that Φ|_b Ω = h. Supported by the MURST project “Geometric Properties of Real and Complex Manifolds”.     

Almost convex groups and the eight geometries

IfM is a closed Nil geometry 3-manifold then ₁(M) is almost convex with respect to a fairly simple geometric generating set. IfG is a central extension or a extension of a word hyperbolic group, thenG is also almost convex with respect to some generating set. Combining these with previously known results shows that ifM is a closed 3-manifold with one of Thurston's eight geometries, ₁(M) is almost convex with respect to some generating set if and only if the geometry in question is not Sol.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

On the structure of singular sets of convex functions

Whenf is a convex function ofR ^h, andk is an integer with 0<k, then the set ^k (f)=x:dim(f(x)k may be covered by countably many manifolds of dimensionh–k and classC ² except an _h–k negligible subset.The author is supported by INdAM     

q‐pseudoconvex and q‐holomorphically convex domains

In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q‐pseudoconvexity and q‐holomorphic convexity: we prove that any open subset $\mathrm{\Omega }?{\mathbb{C}}^n$ with smooth boundary, strictly q‐pseudoconvex, is $(q+1)$‐holomorphically convex; moreover, assuming that Ω verifies an additional assumption, we prove that it is q‐holomorphically convex. We also prove that any open subset of ${\mathbb{C}}^n$ is n‐holomorphically convex.     

Extension of bounded holomorphic functions¶in convex domains

The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson–Andersson type are constructed and estimated for this purpose. Received: 7 July 2000     

Two-dimensional regularity and exactness

We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits [12] introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel–quotient factorisation, extending earlier work of Street and others  and .     

Cogenerators for convex spaces

We give cogenerators for the categories of convex (= finitely superconvex), finitely positively convex, and absolute convex (= finitely totally convex) spaces introduced by Pumplün and Röhrl.Dedicated to our academic teacher Dieter Pumplün on the occasion of his sixtieth birthday.     

Kinematic formulas for finite lattices

In analogy to valuation characterizations and kinematic formulas of convex geometry, we develop a combinatorial theory of invariant valuations and kinematic formulas for finite lattices. Combinatorial kinematic formulas are shown to have application to some probabilistic questions, leading in turn to polynomial identities for Möbius functions and Whitney numbers.     

On the second differentiability of convex surfaces

Properties of pointwise second differentiability of real-valued convex functions in ⁿ are studied. Some proofs of the Busemann-Feller-Aleksandrov theorem are reviewed and a new proof of this theorem is presented.     

On a convex operator for finite sets

Let S be a finite set with m elements in a real linear space and let J_S be a set of m intervals in R. We introduce a convex operator co(S,J_S) which generalizes the familiar concepts of the convex hull, , and the affine hull, , of S. We prove that each homothet of that is contained in can be obtained using this operator. A variety of convex subsets of with interesting combinatorial properties can also be obtained. For example, this operator can assign a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For two types of families J_S we give two different upper bounds for the number of vertices of the polytopes produced as co(S,J_S). Our motivation comes from a recent improvement of the well-known Gauss-Lucas theorem. It turns out that a particular convex set co(S,J_S) plays a central role in this improvement.     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).