On simultaneous inner and outer approximation of shapes
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Journal of Algorithms
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Separation and approximation of polyhedral objects
Computational Geometry: Theory and Applications
Fixed-dimensional linear programming queries made easy
Proceedings of the twelfth annual symposium on Computational geometry
Linear programming queries revisited
Proceedings of the sixteenth annual symposium on Computational geometry
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
Space-time tradeoffs for approximate nearest neighbor searching
Journal of the ACM (JACM)
Proceedings of the twenty-sixth annual symposium on Computational geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Approximate polytope membership queries
Proceedings of the forty-third annual ACM symposium on Theory of computing
Optimal area-sensitive bounds for polytope approximation
Proceedings of the twenty-eighth annual symposium on Computational geometry
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The problem of approximating convex bodies by polytopes is an important and well studied problem. Given a convex body K in Rd, the objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error ε. Results to date have been of two types. The first type assumes that K is smooth, and bounds hold in the limit as ε tends to zero. The second type requires no such assumptions. The latter type includes the well known results of Dudley (1974) and Bronshteyn and Ivanov (1976), which show that in spaces of fixed dimension, O((diam(K)/ε)(d−1)/2) vertices (alt., facets) suffice. Our results are of this latter type. In our first result, under the assumption that the width of the body in any direction is at least ε, we strengthen the above bound to [EQUATION]. This is never worse than the previous bound (by more than logarithmic factors) and may be significantly better for skinny bodies. Our analysis exploits an interesting analogy with a classical concept from the theory of convexity, called the Mahler volume. This is a dimensionless quantity that involves the product of the volumes of a convex body and its polar dual. In our second result, we apply the same machinery to improve upon the best known bounds for answering ε-approximate polytope membership queries. Given a convex polytope P defined as the intersection of halfspaces, such a query determines whether a query point q lies inside or outside P, but may return either answer if q's distance from P's boundary is at most ε. We show that, without increasing storage, it is possible to reduce the best known search times for ε-approximate polytope membership significantly. This further implies improvements to the best known search times for approximate nearest neighbor searching in spaces of fixed dimension.