A Duality between Small-Face Problems in Arrangements of Lines and Heilbronn-Type Problems

Arrangements of lines in the plane and algorithms for computing extreme features of arrangements are a major topic in computational geometry. Theoretical bounds on the size of these features are also of great interest. Heilbronn’s triangle problem is one of the famous problems in discrete geometry. In this paper we show a duality between extreme (small) face problems in line arrangements (bounded in the unit square) and Heilbronn-type problems. We obtain lower and upper combinatorial bounds (some are tight) for some of these problems.