Approximation of Quadrilaterals by Triangles with Respect to Minimal Width

In this paper we prove the following result: Let Q be a quadrilateral which has minimal width w(Q), or simply width, equal to 1. Then there exists a triangle T inscribed in Q such that w(T)≥32≈.866.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(T)\ge \frac{\sqrt{3}}{2} \approx .866.$$\end{document} Moreover, if Q has a center of symmetry then w(T)≥32cos(π/12)≈.8965\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(T)\ge \frac{\sqrt{3}}{2\cos (\pi /12)}\approx .8965$$\end{document}, with equality if and only if Q is a square with side of length 1.