Spiral connectedness of the sections and projections of ℂ-convex sets

Abstract

ABSTRACT. The notion of structural dimension of C-convex sets is introduced. The spiral connectedness of sections and projections of these sets, as well as of the complements of these sections and projections is established. Examples refining L. A. Aizenberg's well-known conjecture about the approximation of strongly linearly convex sets are presented.C-convexity. The complez projeetire space CP n is defined as the set of linear one-dimensional subspaces in the linear space C". We shall denote by p the canonical mapping of C n+l onto CF n . A similar mapping of the space (Cn+l) * of linear functionais on C '~+I onto the dual projecti~e space CP n* will be denoted by the same letter.The standard rule assigns a one-to-one orthogonality relation between linear subspaces of complementary dimension in C n+l and (Cn+1) * to the complex bilinear form (z, w) = w(z) on C "+1 • (C"+I) * . This ylelds the familiar duality correspondence p(l) = p(l "L) between planes in CP" and CP"* whose dimensions add up to n -1.In particular, a hyperplane ~" is canonical[y assigned to any point z E CP n and vice versa, which makes it possible to think of hyperplanes as points of the dual space. (an open or compact set on the complex projective line is said to be acyclic if both the set and its complement are nonempty and connected). This property turned out to be such a natural complex analogue of ordinary convexity that Swedish mathematicians coined a special short term for it, C-con~e,.ity [4], [8].All convex domains and compact sets in C n , as well as all bounded linearly convex domains with smooth boundary [9], are C-convex. Like linear convexity, C-convexity is preserved under automorphisms of CF".Geometric properties of C-convex sets are very difficult to study. In particular, it was only recently that an example of a C-convex domain whose geometry substantially differs from that of ordinary convex domains was discovered [10]: the (2n -1)-dimensional Hansdorff measure of its boundary is infinite, whereas the domain itself is bounded.The well-known problem of approximation by domains with smooth boundaries was posed by L. A. Aizenberg [11] more than fifteen years ago, but no progress was made in its solution since then. Aizenberg's

The Closure and the Interior of C-convex Sets

“…It is well known [30] that the sections of bounded C-convex sets cannot be arbitrary acyclic domains. They always satisfy the additional condition of spiral connection.…”

Peak functions in $$\mathbb {C}$$ C -convex domains

“…Certainly, our results above give also the fatness of the image of a projection of the strictly C-convex normal domain. That kind of property is another one projections of C-convex domains can have (see the results on spiral connectedness in, e.g., Corollary 2.6.7 in [2] or [15]).…”

An example of a bounded $\mathsf C$-convex domain which is not biholomorphic to a convex domain

“…Remarks. (i) In virtue of Proposition 3, we claim that one may conjecture more than Question (a) (see [15]), namely, any C-convex domain containing no complex hyperplanes can be exhausted by bounded C 2 -smooth C-convex domains (this is not true in general without the above assumption); then the Carathéodory pseudodistance and Lempert function will coincide on any C-convex domain.…”