This paper deals with a problem from computational convexity and its application to computer algebra. This paper determines the complexity of computing the Minkowski sum of k convex polytopes in which arc presented either in terms of vertices or in terms of facets. In particular, if the dimension d is fixed, the authors obtain a polynomial time algorithm for adding k polytopcs with up to n vertices. The second part of this paper introduces dynamic versions of Buchbcrgcr's Gr6bncr bases algorithm for polynomial ideals. Using the Minkowski addition of Newton polytopcs, the authors show that the following problem can be solved in polynomial time for any finite set of polynomials T C K [xl,..., Xd], where d is fixed: Does there exist a term ordersuch that T is a Gr6bncr basis for its ideal with respect to -? If not, find an optimal term order for T with respect to a natural Hilbcrt function criterion.

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IntroductionThis paper is the second part of a broader survey of computational convexity, an area of mathematics that has crystallized around a variety of results, problems and applications involving interactions among convex geometry, mathematical programming and computer science. The first part [GrK94a] discussed containment problems. This second part is concerned with computing volumes and mixed volumes of convex polytopes and more general convex bodies. In order to keep the paper self-contained we repeat some aspects that have already been mentioned in [GrK94a]. However, this overlap is limited to Section 1. For further background material and references, see [GrK94a], and for other parts of the survey see [GrK94b] and [GrK94c].Our section headings are as follows.

Characterizing the grain structure of polycrystalline material is an important task in material science. The present paper introduces the concept of generalized balanced power diagrams as a concise alternative to voxelated mappings. Here, each grain is represented by (measured approximations of) its center-of-mass position, its volume and, if available, by its second-order moments (in the non-equiaxed case). Such parameters may be obtained from 3D x-ray diffraction. As the exact global optimum of our model results from the solution of a suitable linear program it can be computed quite efficiently. Based on verified real-world measurements we show that from the few parameters per grain (3, respectively 6 in 2D and 4, respectively 10 in 3D) we obtain excellent representations of both equiaxed and non-equiaxed structures. Hence our approach seems to capture the physical principles governing the forming of such polycrystals in the underlying process quite well.

This paper gives various (positive and negative) results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension. On the negative side, we present several #P-hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes. We show that computing the volume of zonotopes is #P-hard (while each corresponding mixed volume can be computed easily) but also give examples showing that computing mixed volumes is hard even when computing the volume is easy. On the positive side, we derive a randomized algorithm for computing the mixed volumes V (