A further generalization of the colourful Carathéodory theorem
Résumé
Given $d+1$ sets, or colours, $\S_1, \S_2,\ldots,\S_{d+1}$ of points in $\R^d$, a {\em colourful} set is a set $S\subset\bigcup_i\S_i$ such that $|S\cap\S_i|\leq 1$ for $i=1,\ldots,d+1$. The convex hull of a colourful set $S$ is called a {\em colourful simplex}. {\bara}'s colourful {\cara} theorem asserts that if the origin {\zero} is contained in the convex hull of $\S_i$ for $i=1,\ldots,d+1$, then there exists a colourful simplex containing {\zero}. The sufficient condition for the existence of a colourful simplex containing {\zero} was generalized to {\zero} being contained in the convex hull of $\S_i\cup\S_j$ for $1\leq i< j \leq d+1$ by Arocha et al. and by Holmsen et al. We further strengthen the theorem by showing that a colourful simplex containing {\zero} exists if, for $1\leq i< j \leq d+1$, there exists $k\notin\{i,j\}$ such that, for all $x_k\in\S_k$, the convex hull of $\S_i\cup\S_j$ intersects the ray $\overrightarrow{x_k\zero}$ in a point distinct from $x_k$. A slightly stronger version of this new colourful {\cara} theorem is also given. This result provides a short and geometric proof of the previous generalization of the colourful {\cara} theorem. We also give an algorithm to find a colourful simplex containing {\zero} under the strengthened condition. In the plane an alternative and more general proof using graphs is given. In addition, we observe that, in general, the existence of one colourful simplex containing {\zero} implies the existence of at least $\min_i|\S_i|$ colourful simplices containing {\zero}. In other words, any condition implying the existence of a colourful simplex containing {\zero} actually implies the existence of $\min_i|\S_i|$ such simplices.
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