Bounded symmetric domains and polynomial convexity

For every bounded symmetric domain D=G/K in a complex vector space E of finite dimension we determine all polynomially convex compact subsets of E that are invariant under the compact linear group K, where D is realized in the standard way as circular convex domain in E.Mathematics Subject Classification (2000): 17C50, 32E20, 32M15, 32V25     

Balanced metrics on Hartogs domains

An n-dimensional strictly pseudoconvex Hartogs domain D _F can be equipped with a natural Kähler metric g _F . In this paper we prove that if m ₀ g _F is balanced for a given positive integer m ₀ then m ₀>n and (D _F ,g _F ) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space.     

Rectangular convexity of convex domains of constant width

An E R ² is r-convex if for every x, y E there exists a closed rectangle R such that x, y R and R E. Several results about r-convexity appeared in [1]. Its authors formulated a conjecture about conditions for a compact, convex set in R ² to be r-convex. We prove this conjecture in the case of convex domains of constant width.     

On the meromorphic convexity of normality domains in a Stein manifold

Let X be a Stein manifold. Then we prove that for any family ℱ⊂?(X) the normality domain Dℱ) is a meromorphically ?(X)-convex open set of X. Received: 4 November 1999     

A characterization of symmetric tube domains by convexity of Cayley transform images

In this paper, we show that a homogeneous tube domain is symmetric if and only if its Cayley transform image as well as the dual Cayley transform image of the dual tube domain is convex. In this case, the parameters of these Cayley transforms reduce to specific ones, so that they are essentially the usual Cayley transforms defined in terms of Jordan algebra structure.     

A generalization of trigonometric convexity and its relation to positive harmonic functions in homogeneous domains

We consider functions which are subfunctions with respect to the differential operator

Category-measure duality: convexity,midpoint convexity and Berz sublinearity

Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over \({\mathbb {Q}}\) to prove that the graph of a measurable sublinear function that is \({\mathbb {Q}}_{+}\)-homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz’s theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz’s result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein–Doetsch theorem and related continuity results, allowing our conditions to be ‘local’—holding off some exceptional set.     

Toll convexity

A walk $W$ between two non-adjacent vertices in a graph $G$ is called tolled if the first vertex of $W$ is among vertices from $W$ adjacent only to the second vertex of $W$, and the last vertex of $W$ is among vertices from $W$ adjacent only to the second-last vertex of $W$. In the resulting interval convexity, a set $S\subset V\left(G\right)$ is toll convex if for any two non-adjacent vertices $x,y\in S$ any vertex in a tolled walk between $x$ and $y$ is also in $S$. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski–Krein–Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.     

Representation theory and convexity

IfS=G Exp (iW) is a complex open Ol'shanskiî semigroup, whereW is an open elliptic cone, then we considerG-biinvariant domainsD=G Exp (iD_g)S. First we show that the representation ofG×G on eachG-biinvariant irreducible reproducing kernel Hilbert space in Hol(D) is a highest weight representation whose kernel is the character of a highest weight representation ofG. In the second part of the paper we explain how to construct biinvariant Kähler structures on biinvariant Stein domains and show by a certain Legendre transform that the so obtained symplectic manifolds are isomorphic to domains in the cotangent bundleT^*(G).     

Approximate convexity and submonotonicity

It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C¹ functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C¹ functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex.     

Quadratic convexity

We say that a setA ⊂ ℂ ⁿ is quadratically convex if its complement is a union of quadratic hypersurfaces. Some geometric properties of quadratically convex sets are investigated; in particular, they are related to lineally convex sets in a space of higher dimension. We say thatA is strongly quadratically convex if a certain generalization of the Fantappiè transform is surjective, which in effect means that we have a representation for any function holomorphic onA as a superposition of reciprocals of quadratic expressions. The main theorem in this paper gives a sufficient condition for a compact set to be strongly quadratically convex. Using integral representation formulas for holomorphic functions, an explicit inversion formula for the transform is obtained.     

Matrix convexity

The notion of a convex set is generalized. In the definition of ordinary convexity, sums of products of vectors and numbers are used. In the generalization considered in this paper, the role of numbers is played by matrices; this is why we call it matrix convexity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 64–69, January, 1995.     

Random strict convexity and random uniform convexity in random normed modules

Based on the analysis of stratification structure on random normed modules, we first present random strict convexity and random uniform convexity in random normed modules. Then, we establish their respective relations to classical strict and uniform convexity: in the process some known important results concerning strict convexity and uniform convexity of Lebesgue-Bochner function spaces can be obtained as a special case of our results. Further, we also give their important applications to the theory of random conjugate spaces as well as best approximation. Finally, we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces.     

Partial convexity

We consider a generalization of the classical notion of convexity, which is calledpartial convexity. LetV ∋ ℝ ⁿ be some set of directions. A setX ∋ ℝ ⁿ is calledV- convex if the intersection of any line parallel to a vector inV withX is connected. Semispaces and the problem of the least intersection base for partial convexity is investigated. The cone of convexity directions is described for a closed set in ℝ ⁿ . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 406–413, September, 1996. The authors wish to express their gratitude to V. V. Gorokhovik and E. A. Barabanov for useful remarks and discussions. This research was supported in part by the Belorussian Foundation for Basic Research under grant No. F95-016.     

Generalized convexity and inequalities

Let R₊=(0,∞) and let M be the family of all mean values of two numbers in R₊ (some examples are the arithmetic, geometric, and harmonic means). Given m₁,m₂∈M, we say that a function is (m₁,m₂)-convex if f(m₁(x,y))?m₂(f(x),f(y)) for all x,y∈R₊. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m₁,m₂)-convexity on m₁ and m₂ and give sufficient conditions for (m₁,m₂)-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.