Carathéodory, Helly and the others in the max-plus world

Δ-凸性中的Carathéodory数与交换数

Carath$é$odory Number and Exchange Number in $Δ$-convexity

本文研究Δ-凸性中Carathéodory数与交换数，针对多种图族及笛卡尔积、强积、字典序积等标准图乘积展开讨论。

Formulas and Upper Bounds for the Carath{é}odory Number of Hamming Graphs

Let $G$ be a simple graph and let $S$ be a subset of its vertices. We say that $S$ is $P_3$-convex if every vertex $v \in V(G)$ that has at least two neighbors in $S$ also belongs to $S$. The $P_3$-hull set of $S$ is the smallest $P_3$-convex set of $G$ that contains $S$. Carathéodory number of a graph $G$, denoted by $c(G)$, is the smallest integer $c$ such that for every subset $S \subseteq V(G)$ and every vertex $p$ in the $P_3$-hull of $S$, there exists a subset $F \subseteq S$ with $|F| \leq c$ such that $p$ belongs to the $P_3$-hull of $F$. In this article, we present upper bounds and fo…

Quantitative Tverberg, Helly, & Carathéodory theorems

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc.

Helly-type Problems

In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems, and explain some of its main themes and goals.

Theorems of Carathéodory, Helly, and Tverberg without dimension

We prove a no-dimensional version of Carathédory's theorem: given an $n$-element set $P\subset \Re^d$, a point $a \in \conv P$, and an integer $r\le d$, $r \le n$, there is a subset $Q\subset P$ of $r$ elements such that the distance between $a$ and $\conv Q$ is less than $\diam P/\sqrt {2r}$. A general no-dimension Helly type result is also proved with colourful and fractional consequences. Similar versions of Tverberg's theorem and some of their extensions are also established.