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We present a newpiv ot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:(a) No additional storage is required beyond the input data; (b) The output list produced is free of duplicates; (c) The algorithm is extremely simple, requires no data structures, and handles all degenerate cases; (d) The running time is output sensitive for non-degenerate inputs; (e) The algorithm is easy to efficiently parallelize.Forexample, the algorithm finds the v vertices of a polyhedron in R d defined by a nondegenerate system of n inequalities (or dually,the v facets of the convex hull of n points in R d ,w here each facet contains exactly d givenp oints) in time O(ndv)a nd O(nd) space. The v vertices in a simple arrangement of n hyperplanes in R d can be found in O(n 2 dv)t ime and O(nd)s pace complexity.T he algorithm is based on inverting finite pivotalgorithms for linear programming.-2-

We study several known volume computation algorithms for convex d-polytopes by classifying them into two classes, triangulation methods and signed-decomposition methods. By incorporating the detection of simplicial faces and a storing/reusing scheme for face volumes we propose practical and theoretical improvements for two of the algorithms. Finally we present a hybrid method combining advantages from the two algorithmic classes. The behaviour of the algorithms is theoretically analysed for hypercubes and practically tested on a wide range of polytopes, where the new hybrid method proves to be superior.