In 1935 ErdÅ‘s and Szekeres proved that for any integer n â‰¥ 3 there exists a smallest positive integer N(n) such that any set of at least N(n) points in general position in the plane contains n pointsâ€¦ (More)

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 Gohbergâ€“Markusâ€“Hadwigerâ€¦ (More)

This is a survey of known results and still open problems on antipodal properties of finite sets in Euclidean space. The exposition follows historical lines and takes into consideration both metricâ€¦ (More)

Nonempty sets X1 and X2 in the Euclidean space R are called homothetic provided X1 = z+Î»X2 for a suitable point z âˆˆ R and a scalar Î» 6= 0, not necessarily positive. Extending results of SÃ¼ss andâ€¦ (More)

There are determined sharp lower bounds for the number of shadowâ€“ boundaries, and illuminated regions of a convex body in En, exhibiting extremal properties of the simplex and the parallelotope,â€¦ (More)

Let F = fC 1 ; C 2 ; : : : ; C n g be a family of n disjoint convex bodies in the plane. We say that a set V of exterior light sources illuminates F, if for every boundary point u of any member of Fâ€¦ (More)

Let SH denote the homothety class generated by a convex set S âŠ‚ R: SH = {a + Î»S | a âˆˆ R, Î» > 0}. We determine conditions for the Minkowski sum BH + CH or the Minkowski difference BH âˆ¼ CH of homothetyâ€¦ (More)

In 1964 GrÃ¼nbaum conjectured that any primitive set illuminating from within a convex body inEd, d â‰¥ 3, has at most 2 d points. This was confirmed by V. Soltan in 1995 for the cased = 3. Here we giveâ€¦ (More)

A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discreteâ€¦ (More)