On the area of planar convex sets containing many lattice points

Inequalities for Lattice Constrained Planar Convex Sets

Every convex set in the plane gives rise to geometric functionals such as the area, perimeter, diameter, width, inradius and circumradius. In this paper, we prove new inequalities involving these geometric functionals for planar convex sets containing zero or one interior lattice point. We also conjecture two results concerning sets containing one interior lattice point. Finally, we summarize k...

Random Walks on Nite Convex Sets of Lattice Points

On the Odd Area of Planar Sets

The main result in the paper is a construction of a simple (in fact, just a union of two squares) set T in the plane with the following property. For every > 0 there is a family F of an odd number of translates of T such that the area of those points in the plane that belong to an odd number of sets in F is smaller than .

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

On the Boundary of the Union of Planar Convex Sets

We give two alternative proofs leading to di erent generalizations of the following theorem of [1]. Given n convex sets in the plane, such that the boundaries of each pair of sets cross at most twice, then the boundary of their union consists of at most 6n 12 arcs. (An arc is a connected piece of the boundary of one of the sets.) In the generalizations we allow pairs of boundaries to cross more...