2007/10/25 by W. R. G. James, James, W. R. G., Iwan Jensen +5
Computer Science · Engineering · Mathematics · #05A15 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Optimization and Packing Problems #Point processes and geometric inequalities #math.CO #msc:05A15
paper · pdf · doi:10.48550/arxiv.0710.4606
53 pages
arxiv created 2007/10/25 · openalex publication_date 2007/10/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', m. Such polygons are called m-convex polygons and are characterised by having up to m indentations in the side. We use a `divide and conquer' approach, factorising 2-convex polygons by extending a line along the base of its indents. We then use the inclusion-exclusion principle, the Hadamard product and extensions to known methods to derive the generating functions for each case.