Approximation of quadrilaterals by triangles with respect to minimal width (Q6045113)

There are many open problems concerning approximation of convex bodies by polytopes. One such problem is the following: \textbf{Problem} [\textit{P. Brass} et al., Research problems in discrete geometry. New York, NY: Springer (2005; Zbl 1086.52001)]. What is the largest number \(c_n\) such that every plane convex set \(K\) has an inscribed \(n\)-gon \(P_n\) with \(w(P_n)\geq c_n.w(K)\)? where \(w(M)\) denotes the minimum width of a convex set \(M\) ( We may recall that the width of a given convex set \(K\) in the direction \(u \in S^1\), \(w(u)\), is the distance between the two supporting lines of \(K\) which are orthogonal to \(u\), and \(w(K)\) is the minimum of these distances, i.e., \(w(K)= \min_{u \in S^1} w(u)\)). \textit{M. Lassak} [Colloq. Math. 149, No. 1, 21--32 (2017; Zbl 1421.52007)] proved that for every convex body \(K\) of width \(1\), there is a triangle \(T\) inscribed in \(K\) such that \(w(T) \geq \frac{3-\sqrt{3}}{2}\approx 0.634\). If in addition, \(K\) is a centrally symmetric set, then \(w(T)\geq \frac{3}{4}\) and equality holds if and only if \(K\) is a disk. In this paper, the authors prove the following: Theorem. Let \(Q\) be a quadrilateral with \(w(Q)=1\), then there exist a triangle \(T\) inscribed in \(Q\) such that \(w(T)\geq \frac{\sqrt{3}}{2}\approx .866\). Moreover, if \(Q\) is a parallelogram then \(w(T)\geq \frac{\sqrt{3}}{2\cos(15^{\circ})} \approx .8965\), with equality if and only if \(Q\) is a square with side of length \(1\).