Quantitative Transversal Theorems in the Plane

Hadwiger's theorem is a variant of Helly-type theorems involving common transversals to families of convex sets instead of common intersections. In this paper, we obtain a quantitative version of Hadwiger's theorem on the plane: given an ordered family of pairwise disjoint and compact convex sets in $\mathbb{R}^2$ and any real-valued monotone function on convex subsets of $\mathbb{R}^2,$ if every three sets have a common transversal, respecting the order, such that the intersection of the sets with each half-plane defined by the transversal are valued at least (or at most) some constant $\alpha,$ then the entire family has a common transversal with the same property. Unlike previous generalizations of Hadwiger's theorem, we prove that disjointness is necessary for the quantitative case. We also prove colorful versions of our results

Topology of geometric joins

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carath\'eodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least $d+1$ sets, where $d$ is the dimension of the space, then the geometric join is contractible. We are able to prove this when $d$ equals $2$ and $3$, while for larger $d$ we show that the geometric join is contractible provided the number of sets is quadratic in $d$. We also consider a matroid generalization of geometric joins and provide similar bounds in this case

On Partitioning Colored Points

P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can be separated along the colors. In this paper, we show a more colorful theorem