Relative convex hulls have been suggested for multigridconvergent surface area estimation. Besides the existence of a convergence theorem there is no efficient algorithmic solution so far for calculating 3D relative convex hulls. This article discusses an approximative solution based on minimum-length polygon calculations. It is illustrated that this approximative calculation also proves (exper...

We give a new proof of an isoperimetric inequality for family closed surfaces, which have Gaussian curvature identically equal to one wherever the surface is smooth. These surfaces are formed from convex, spherical polygon, with each vertex polygon leading non-smooth point on surface. For example, lune revolution, two tips. Combined straightforward approximation argument, this was first proved ...

We present an exhaustive search of all families of convex pentagons which tile the plane. This research shows that there are no more than the already 15 known families. In particular, this implies that there is no convex polygon which allows only non-periodic tilings.