Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

Sequential and Parallel Algorithms for Finding a Maximum Convex Polygon

AbstractThis paper investigates the problem where one is given a finite set of n points in the plane each of which is labeled either “positive” or “negative”. We consider bounded convex polygons, the vertices of which are positive points and which do not contain any negative point. It is shown how such a polygon which is maximal with respect to area can be found in time O(n3logn). With the same running time one can also find such a polygon which contains a maximum number of positive points. If, in addition, the number of vertices of the polygon is restricted to be at most M, then the running time becomes O(Mn3logn). It is also shown how to find a maximum convex polygon which contains a given point in time O(n3logn). Two parallel algorithms for the basic problem are also presented. The first one runs in time O(nlogn) using O(n2) processors, the second one has polylogarithmic time but needs O(n7) processors. Instead of using the area or the number of positive points contained in the polygon as the quantity to be maximized one may also use other measures fulfilling a certain additive property, however, this may affect the running time