This 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].

On the Complexity of Some Basic Problems in Computational Convexity

When approx ima te l y 15 years ago we became i n t e r e s t e d in the computat i o n a l geometry o f po lyhedra , we soon r e a l i z e d tha t c e r t a i n bas ic not i o n s (e .g . face o f a po lyhedron) were not p r o p e r l y de f ined , and tha t no a p p r o p r i a t e theory was a v a i l a b l e . Most th ings t h e r e f o r e had to be done on a p u r e l y i n t u i t i v e bas i s , lead ing i n to d i f f i c u l t i e s e s p e c i a l l y in c e r t a i n s i n g u l a r cases ( c f . [ 0 9 ] ) . Never the less a la rge number o f p u b l i c a t i o n s in the f i e l d have s ince appeared, e .g . [ 08 ] , [ 10 ] , [13] to [18] and [21] to mention on l y some of the most s i g n i f i c a n t .

Elementary Set Operations with d-Dimensional Polyhedra

A sweep-plane algorithm by Lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes. In our method an affine hyperplane is swept through the given polytope until a random fraction (sampled from a proper univariate distribution) of the volume of the polytope is covered. Then the intersection of the plane with the polytope is a simple polytope with smaller dimension. In the second part we apply this method to construct a black-box algorithm for log-concave and T-concave multivariate distributions by means of transformed density rejection. (author's abstract)

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A sweep-plane algorithm for generating random tuples in simple polytopes

The Euler Characteristics of Discrete Objects and Discrete Quasi-Objects

Plane-sweep algorithms form a fairly general approach to two-dimensional problems of computational geometry. No corresponding general space-sweep algorithms for geometric problems in 3- space are known. We derive concepts for such space-sweep algorithms that yield an efficient solution to the problem of solving any set operation (union, intersection, ...) of two convex polyhedra. Our solution matches the best known time bound of O(n log n), where n is the combined number of vertices of the two polyhedra.

H. Bieri

Papers

A sweep-plane algorithm for computing the volume of polyhedra represented in boolean form