# Complex Convexity and Analytic Functionals by Mats Andersson

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A set in advanced Euclidean house is termed C-convex if all its intersections with complicated traces are contractible, and it really is stated to be linearly convex if its supplement is a union of advanced hyperplanes. those notions are intermediates among usual geometric convexity and pseudoconvexity. Their value was once first manifested within the pioneering paintings of André Martineau from approximately 40 years in the past. considering the fact that then a good number of new comparable effects were got by means of many alternative mathematicians. the current publication places the fashionable concept of advanced linear convexity on an excellent footing, and provides an intensive and up to date survey of its present prestige. functions comprise the Fantappié transformation of analytic functionals, imperative illustration formulation, polynomial interpolation, and recommendations to linear partial differential equations.

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7 we know that this really is a weaker condition. 9 is mimicked on the usual proof of the Runge theorem in one variable, and the very same argument, but with the affine linear functions 9t replaced by polynomials of arbitrary degrees, was used by Stolzenberg in [1] in order to give the following simple characterization of polynomial convexity for general, not necessarily linearly convex, subsets of en: The compact set E is polynomially convex if and only if through any point in the complement of E there passes an algebraic hypersurface, which can be continuously pulled away to 0 infinity without intersecting E.

_Let U be an arbitrary open neighborhood of E. For any line £ through a we let Uf be the set of points z on £, such that there is a closed curve through a in Une, with respect to which z has index 1. (In other words, we take the component of Un £ that contains a, and "fill in" the holes. ) If we now let U be the union of all the Uf, then U is C-starlike by definition, and we claim that it is in fact an open neighborhood of E. ~ince Ene is cOEnected by assumption, it followsJ;hat Ene c Uf and therefore E C U.

Any weakly linearly convex open set E in C n is pseudoconvex . Proof. Let K be a compact subset of E. The holomorphic hull KE is then a bounded subset of Cn , since it is clearly contained in the ordinary convex hull of K. It is thus enough to show that the closure of KE is contained in E. Let therefore Zo be a boundary point of E. By the weak linear convexity we can find an affine linear function I vanishing at Zo and such that the zero set of I lies entirely ou~ide E. Since sUPK 11/11 is bounded it follows that Zo is not in the closure of K E as desired.