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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.

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