New Views on Some Old Questions of Combinatorial Geometry

Abstract

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 Brunel's (1897) problem about the number of hexagons in 3-valent planar maps with precisely 3 digons; substantial results on Sylvester's (1867) problem about the number of collinear triplets possible with n points; solution and extension of de Rocquigny's (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 Wernicke's (1904) and Kotzig's (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.

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Document Details

Document Type
Technical Report
Publication Date
Sep 01, 1973
Accession Number
AD0768893

Entities

People

  • Branko Grunbaum

Organizations

  • University of Washington

Tags

Communities of Interest

  • Air Platforms

DTIC Thesaurus Topics

  • Algebraic Geometry
  • Analogs
  • Geometry
  • Graph Theory
  • Inequalities
  • Materials
  • Mathematics
  • Military Research
  • New York
  • Polygons
  • Security
  • Sequences
  • Sizes (Dimensions)
  • Thinking
  • Three Dimensional
  • Topology
  • Triangulation

Readers

  • Calculus or Mathematical Analysis
  • Graph Algorithms and Convex Optimization.