The minimum area of a simple polygon with given side lengths

Abstract

Answering a question of H. Harborth, for any given a 1,...,a n > 0, satisfying \(a_i < \sum\limits_{j \ne i} {a_j } \)we determine the infimum of the areas of the simple n-gons in the Euclidean plane, having sides of length a 1,...,a n (in some order). The infimum is attained (in limit) if the polygon degenerates into a certain kind of triangle, plus some parts of zero area. We show the same result for simple polygons on the sphere (of not too great length), and for simple polygons in the hyperbolic plane. Replacing simple n-gons by convex ones, we answer the analogous questions. The infimum is attained also here for degeneration into a certain kind of triangle.