Covering convex sets with non-overlapping polygons (Q912393)

Improving results of \textit{R. Wenger} [Rep. TR-SOCS-86.19, School Comput. Sci., McGill Univ., Montreal, Quebec (1986)] the authors give sharp bounds for the total number of sides and distinct slopes of n non- overlapping convex polygons, which cover \(n\geq 3\) convex, compact and pairwise-disjoint sets in \(E^ 2\). More precisely, these polygons have a total of no more than 6n-9 sides, and with no more than 3n-6 distinct slopes. Examples reaching these bounds are constructed and the connections to two problems of combinatorial geometry, i.e. to a transversal problem and to a triangulation problem, are presented.