Exact computation of the halfspace depth

We suggest a theoretical framework for computing the exact value of the halfspace depth of a point $z$ w.r.t. a data cloud $x_1,\dots,x_n$ of $n$ points in arbitrary dimension. Based on this framework a whole class of algorithms can be derived. In all of these algorithms the depth is calculated as the minimum over a finite number of depth values w.r.t. proper projections of the data cloud. Three variants of this class are studied in more detail. All of these algorithms are capable of dealing with data that are not in general position and even with data that contain ties. As our simulations show, all proposed algorithms prove to be very efficient.