Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.Comment: 27 pages, 10 figure

Lower Bounds for Pinning Lines by Balls

A line L is a transversal to a family F of convex objects in R^d if it intersects every member of F. In this paper we show that for every integer d>2 there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the property that every subfamily of size 2d-2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957

On a colorful problem by Dol'nikov concerning translates of convex bodies

In this note we study a conjecture by Jer\'onimo-Castro, Magazinov and Sober\'on which generalized a question posed by Dol'nikov. Let $F_1,F_2,\dots,F_n$ be families of translates of a convex compact set $K$ in the plane so that each two sets from distinct families intersect. We show that, for some $j$, $\bigcup_{i\neq j}F_i$ can be pierced by at most $4$ points. To do so, we use previous ideas from Gomez-Navarro and Rold\'an-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square

Colourful transversal theorems

We prove the colourful versions of three clasical transversal theorems: The Katchalski-Lewis Theorem "T(3) implies T-k", the "T(3) implies T" Theorem for well distributed sets, and the Goodmann-Pollack Transversal Theorem for hyperplanes

Discrete Geometry

[no abstract available