Leírás
A problem posed by Erdős in 1945 initiated the study of non-separable arrangements of convex bodies. A finite family of convex bodies in Euclidean $d$-space is called a non-separable family (or NS-family) if every hyperplane intersecting their convex hull also meets at least one member of the family. Minimal coverings of NS-families consisting of positive homothetic convex bodies have been explored in several recent studies. In this lecture, we extend these results to weakly non-separable families of convex $d$-polytopes. Moreover, we investigate maximal weakly non-separable families of $d$-cubes and examine stability versions as well as weakly $k$-impassable families of convex $d$-polytopes for $0 < k < d-1$.
Joint work with Z. Lángi.