New Views on Some Old Questions of Combinatorial Geometry
WASHINGTON UNIV SEATTLE
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Several rather old problems in combinatorial geometry have recently been solved, mostly within the frameworks of either the theory of convex polytopes, or that of arrangements of lines or curves. Among those surveyed Solution of Brunels 1897 problem about the number of hexagons in 3-valent planar maps with precisely 3 digons substantial results on Sylvesters 1867 problem about the number of collinear triplets possible with n points solution and extension of de Rocquignys 1897 problem about the number of points of tangency in systems of mutually non-overlappins circles corrections and generalizations of previously published assertions about Venn diagrams 1880. Among the new results established are Sharpenings of Wernickes 1904 and Kotzigs 1955 results about edges with endpoints of low valence in 3- dimensional convex polyhedra, and substantial improvements of previously known results on a problem of Drdos 1962 on collinear multiplets of points.
- Theoretical Mathematics