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We develop algorithms for the approximation of a convex polytope in R 3 by polytopes that are either contained in it or containing it, and that have fewer vertices or facets, respectively. The approximating polytopes achieve the best possible general order of precision in the sense of volume-difference. The running time is linear in the number of vertices… (More)

The problem of the existence of an equi-partition of a curve in R n has recently been raised in the context of computational geometry (see [2] and [3]). The problem is to show that for a (continuous) curve Γ : [0, 1] → R n and for any positive integer N, there exist points t 0 = 0 < t 1 <. .. < t N −1 < 1 = t N , such that d(Γ(t i−1), Γ(t i)) = d(Γ(t i),… (More)

Let P be a convex polytope in R d , d = 3 or 2, with n vertices. We present linear time algorithms for approximating P by simpler polytopes. For instance, one such algorithm selects k < n vertices of P whose convex hull is the approximating polytope. The rate of approximation, in the Hausdorff distance sense, is best possible in the worst case. An analogous… (More)

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