The power of geometric duality
BIT - Ellis Horwood series in artificial intelligence
Algorithms for Polytope Covering and Approximation
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
Building triangulations using ε-nets
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The Effect of Corners on the Complexity of Approximate Range Searching
Discrete & Computational Geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
Polytope approximation and the Mahler volume
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Tight Lower Bounds for Halfspace Range Searching
Discrete & Computational Geometry - Special Issue: 26th Annual Symposium on Computational Geometry; Guest Editor: David Kirkpatrick
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Approximating convex bodies is a fundamental question in geometry and has applications to a wide variety of optimization problems. Given a convex body K in REd for fixed d, the objective is to minimize the number of vertices or facets of an approximating polytope for a given Hausdorff error ε. The best known uniform bound, due to Dudley (1974), shows that O((diam(K)/ε)(d-1)/2) facets suffice. While this bound is optimal in the case of a Euclidean ball, it is far from optimal for skinny convex bodies. We show that, under the assumption that the width of the body in any direction is at least ε, it is possible to approximate a convex body using O(√area(K)/ε(d-1)/2) facets, where area(K) is the surface area of the body. This bound is never worse than the previous bound and may be significantly better for skinny bodies. This bound is provably optimal in the worst case and improves upon our earlier result (which appeared in SODA 2012). Our improved bound arises from a novel approach to sampling points on the boundary of a convex body in order to stab all (dual) caps of a given width. This approach involves the application of an elegant concept from the theory of convex bodies, called Macbeath regions. While Macbeath regions are defined in terms of volume considerations, we show that by applying them to both the original body and its dual, and then combining this with known bounds on the Mahler volume, it is possible to achieve the desired width-based sampling.