Polytope Approximation and the Mahler Volume

The problem of approximating convex bodies by polytopes is an important and well studied problem. Given a convex body K in R d, 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 Õ(√area(K)/ε (d-1)/2). 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. Copyright © SIAM