We propose a notion of depth with respect to a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ which we call $\text{dep}_\mathcal{F}$. We begin showing that $\text{dep}_\mathcal{F}$ satisfies some expected properties for a measure of depth and that this definition is closely related to the notion of depth proposed by J. Tukey. We show that some properties of Tukey depth extend to $\text{dep}_\mathcal{F}$ and we point out some key differences.
We then focus on the following centerpoint-type question: what is the best depth $α_{d,k}$ that we can guarantee under the hypothesis that the family $\mathcal{F}$ is $k$-intersecting? We show a key connection between this problem and a purely combinatorial problem on hitting sets. The relationship is useful in both directions. On the one hand, for values of $k$ close to $d$ the combinatorial interpretation gives a good bound for $k$. On the other hand, for low values of $k$ we can use the classic Rado's centerpoint theorem to get combinatorial results of independent interest. For intermediate values of $k$ we present a probabilistic framework to improve the bounds and illustrate its use in the case $k\approx d/2$. These results can be though of as an interpolation between Helly's theorem and Rado's centerpoint theorem.
As an application of these results we find a Helly-type theorem for fractional hyperplane transversals. We also give an alternative and simpler proof for a transversal result of A. Holmsen.
We prove that every finite family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by $9$ points. This improves the bound of $13$ proved by Gyárfás, Kleitman, and Tóth in 2001.
For $n\leq d$, a family ${\cal F}=\{C_0,C_1,\ldots, C_n\}$ of compact convex sets in $R^d$ is called an $n$-critical family provided any $n$ members of ${\cal F}$ have a non-empty intersection, but $\bigcap_{i=0}^n C_i=\varnothing$. If $n=d$ then a lemma on the intersection of convex sets due to Klee implies that the $d+1$ members of the $d$-critical family enclose a `hollow' in $R^d$, a bounded connected component of $R^d\setminus\bigcup_{i=0}^n C_i.$ Here we prove that the closure of the convex hull of a hollow in $R^d$ is a $d$-simplex.
In 2012, a family of benzenoids was introduced by Cruz, Gutman, and Rada, which they called convex benzenoids. In this paper we introduce the convexity deficit, a new topological index intended for benzenoids and, more generally, fusenes. This index measures by how much a given fusene departs from convexity. It is defined in terms of the boundary-edges code. In particular, convex benzenoids are exactly the benzenoids having convexity deficit equal to 0. Quasi-convex benzenoids form the family of non-convex benzenoids that are closest to convex, i.e., they have convexity deficit equal to 1. Fin…
A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, let $\mathcal{I}^{(r)}_v(G)$ denote the family of independent sets of size $r$ that contain $v$. This family is called an $r$-star. Then $G$ is said to be $r$-EKR if no intersecting subfamily of $ \mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star. Let $k, n, r \geq 1$, and let $T(n, k)$ be the tree of depth two in which the root has degree $n$ and every neighbou…
The extremal functions $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ are linear in $n$, the latter posed by Braß-Károlyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$. We give a forbidden subgraph characterization for a family $\cal T$ of ordered trees with $k$ edges, and show that $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in …
According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$ points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of $n$ convex bodies in the plane is said to be in convex position if the convex hull of the union of no $n-1$ of its members contains the remaining one. If any thre…
We define a $C(k)$ to be a family of $k$ sets $F_1,\dots,F_k$ such that $\textrm{conv}(F_i\cup F_{i+1})\cap \textrm{conv}(F_j\cup F_{j+1})=\emptyset$ when $\{i,i+1\}\cap \{j,j+1\}=\emptyset$ (indices are taken modulo $k$). We show that if $\mathcal{F}$ is a family of compact, convex sets that does not contain a $C(k)$, then there are $k-2$ lines that pierce $\mathcal{F}$. Additionally, we give an example of a family of compact, convex sets that contains no $C(k)$ and cannot be pierced by $\left\lceil \frac{k}{2} \right\rceil -1$ lines.
We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colo…
