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Polytope Approximation and the Mahler Volume (Preprint)

机译：多面体逼近和马勒体积（预印本）

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摘要

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 epsilon. Results to date have been of two types. The first type assumes that K is smooth, and bounds hold in the limit as epsilon 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)=epsilon)(d-1)/2) vertices (alt., facets) suffice. Our results are of this latter type.

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