By Mats Andersson

A set in complicated Euclidean house is termed **C**-convex if all its intersections with complicated strains are contractible, and it truly is stated to be linearly convex if its supplement is a union of complicated hyperplanes. those notions are intermediates among usual geometric convexity and pseudoconvexity. Their significance was once first manifested within the pioneering paintings of André Martineau from approximately 40 years in the past. considering that then a number of new similar effects were bought through many various mathematicians. the current ebook places the fashionable thought of complicated linear convexity on a pretty good footing, and offers a radical and up to date survey of its present prestige. functions contain the Fantappié transformation of analytic functionals, vital illustration formulation, polynomial interpolation, and recommendations to linear partial differential equations.

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**Example text**

The next example shows that there exist non-degenerate open C-convex sets which are not bounded in any affinization. 14. 8. Let H be a real hyperplane in cn, and let E be a non-degenerate C-convex compact subset of H. Then E is in fact convex. Indeed, take two points a and b in E. If the complex line through a and b is not contained in H then its intersection with H is a real line, and it follows that the segment [a, b] is contained in E. If on the other hand the complex line through a and b is contained in H, then there is also a unique complex hyperplane H' C H containing a and b.

To exhibit such a point one may proceed as follows: Choose any two points Zl and Z2 in E 1 , such that the line segment a = [Zl, Z2] is not contained in E 1 . Then join Zl and Z2 by a piecewise linear curve --y inside E 1 . After possibly replacing Zl and Z2 by two other points in --yna, one can assume that the curve --y does not intersect the segment a except at its endpoints. Let ~ denote the compact polygon bounded by --y and a, and consider the set K of points in ~ n eEl at maximal distance from a.

1. A subset E of IP' is said to be C-convex if E i= IP and both E and its complement IP \ E are connected. A subset E of IP'n is called C-convex if all its intersections with complex lines are C-convex. A C-convex set which is not contained in any complex hyperplane and which does not contain any affine complex line will be called non-degenerate. 26 Chapter 2. Complex Convexity It is clear that projective bijections preserve ((>convexity, for they map lines to lines and connected sets to connected sets.