In this paper we summarize the known results and the main tools concerning complete systems of inequalities for families of convex sets. We discuss also the possibility of using these systems to determine particular subfamilies of planar convex sets with specific geometric significance. We also analyze complete systems of inequalities for 3-rotationally… (More)
We give a generalization of Bender's area-perimeter relation for plane lattice-point-free convex regions to simply connected regions, thus we solve a problem posed by M. Silver 10]. Further the result is used for a lattice version of the Dido problem.
Let K ⊂ R 2 be a compact convex set in the plane. A halving chord of K is a line segment pˆp, p, ˆ p ∈ ∂K, which divides the area of K into two equal parts. For every direction v there exists exactly one halving chord. Its length h A (v) is the corresponding (area) halving distance. In this article we give inequalities relating the minimum and maximum… (More)
Let E be a subset of a convex, open, bounded, planar set G. Let P (E, G) be the relative perimeter of E (the length of the boundary of E contained in G). We obtain relative geometric inequalities comparing the relative perimeter of E with the relative diameter of E and with its relative inradius. We prove the existence of both extremal sets and maximizers… (More)