Acute triangulations of double planar convex bodies (Q2915423)

Let \(K\) be a planar convex body having the coordinate axes as axes of symmetry and with boundary \(\operatorname{bd}K\) of class \(C^2\). Let \(a,b\) be the points of \(\operatorname{bd}K\) on the positive coordinate semi-axes and \(c=\min\bigl\{\| a\| ,\| b\| \bigr\}\). This note is devoted to prove that if the curvature of the curve \(\operatorname{bd}K\cap\bigl\{(x,y):x,y\geq 0\bigr\}\) is either monotone or bounded from above by \(2/c\), then the double convex body \(2K\) admits an acute (geodesic) triangulation with 72 triangles.NEWLINENEWLINE Here a (geodesic) triangulation is called acute if the angles of all (geodesic) triangles are smaller than \(\pi/2\), whereas the double convex body \(2K\) is surface homeomorphic to the circle, consisting of two planar isometric convex bodies \(K\) and \(K'\), with boundaries glued according to the isometry.