The Volume of the Convex Hull of a Body and its Homothetic Copies

Survey Paper Combinatorial problems on the illumination of convex bodies

This is a review of various problems and results on the illumination of convex bodies in the spirit of combinatorial geometry. The topics under review are: history of the Gohberg–Markus–Hadwiger problem on the minimum number of exterior sources illuminating a convex body, including the discussion of its equivalent forms like the minimum number of homothetic copies covering the body; generalizat...

Sweep Line Algorithm for Convex Hull Revisited

Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...

Covering a Convex Body by Its Negative Homothetic Copies

L-CONVEX SYSTEMS AND THE CATEGORICAL ISOMORPHISM TO SCOTT-HULL OPERATORS

The concepts of $L$-convex systems and Scott-hull spaces are proposed on frame-valued setting. Also, we establish the categorical isomorphism between  $L$-convex systems and Scott-hull spaces. Moreover, it is proved that the category of $L$-convex structures is  bireflective in the category of $L$-convex systems. Furthermore, the quotient systems of $L$-convex systems are studied.

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 $k$ ...