Abstract: Consider n points in general position in d-dimensional space. For a k-element subset the degree is the number of empty simplices with this k-set as base. The k-degree of the set of n points is defined as the maximum degree over all k-element subset. An old (and still unsolved) question by Erdös asks whether the 2-degree of a planar point sets is unbounded as n tends to infinity. We investigate the degree of random point sets consisting…

Let $d \ge 2$. Let $A$ be a {\em random matrix} taking values in one of the following (semi-)groups: \begin{itemize}     \item $GL(d,\mathds{R})$, the group of invertible real matrices     \item $\R_> \times O(d)$, the group of similarity matrices     \item $ M(d \times d, \mathds{R}_\ge)$, the semi-group of nonnegative matrices \end{itemize} \medskip \textbf{Goal}: To study properties of the {\em left} random walk (=product of matrices) $$…