Stellar discriminants and equipartitions (Q452105)

Let \(\Delta\) be a \(d\)-simplex embedded in some Euclidean space \(\mathbb{R}^m\), and let \(S\) be a set of \(n\) points in the interior of \(\Delta\). The author investigates the problem of finding a stellar subdivision of \(\Delta\) such that the interiors of all the \(d\)-simplices of the subdivision contain the same number of points of \(S\). He proves that, if the points are in a kind of general position, then such a subdivision always exists and he presents an algorithm (with complexity \(O(n^2)\)) to find the center of the stellar subdivision. Examples of configurations with no stellar equipartitions are shown. The paper also contains applications of stellar equipartitions in various contexts.