Finding the largest inscribed axially symmetric polygon for a convex polygon

Abstract

A planar figure F is axially-symmetric if there is a line l such that reflection around l leaves F invariant. We consider the following problem: Given a compact convex figure C in the plane of area one, what is the area γ(C) of the largest axially-symmetric figure F inscribed in C? We are interested in the worst case, that is, in γ := inf_Cγ(C), where the infimum is taken over all compact convex figures C of area one in the plane. Lassak proved that γ >= 2/3, that is, for every convex compact figure C of area one, there is an inscribed axially-symmetric figure of area at least 2/3. For the special case that C is a triangle, Buda and Mislow proved that $γ = 2(root{2} - 1)$. Interestingly, Nohl showed exactly the same bound $γ = 2(root{2} - 1)$ for centrally symmetric convex figures C. Both bounds are tight - there is a triangle and a parallelogram of area one that do not admit an axially symmetric subset of area larger than $2(root{2} - 1)$. For general, not centrally symmetric convex figures, there is still a large gap: $2/3 <= γ <= 2(root{2} - 1)$. Since the upper bound is the correct answer for both triangles and centrally-symmetric figures, one might conjecture that it is the correct bound in general. We will show that this conjecture is false: We have performed experiments in which we generated convex polygons in the plane, and then computed the largest inscribed axially-symmetric subset. From our experiments, we conjecture that the true bound is around 0.81, which is less than $2(root{2} - 1)$, but much higher than the known lower bound of 2/3. The worst case seems to arise for convex quadrilaterals. We give an explicit example of a convex quadrilateral that does not admit an axially symmetric subset of area larger than 0.8158382599948. Since this bound is only slightly smaller than the conjecture we are disproving, and since we can only compute the largest axially-symmetric subset approximately, we have to make a considerable effort t...