How often is a polygon bounded by three sides?

Abstract

LetL _n be the set of lines (no two parallel) determining ann-sided bounded faceF in the Euclidean plane. We show that the number,f(L _n), of triples fromL _n that determine a triangle containingF satisfies$n - 2 \leqq f\left( {L_n } \right) \leqq \frac{n}{6}\left[ {\frac{{n^2 - 1}}{4}} \right]$ and these bounds are best. This result is generalized tod-dimensional Euclidean space (without the claim that the upper bound is attainable).