Linear time approximation of 3D convex polytopes

Linear time approximation of 3D convex polytopes

Hausdorff approximation of 3D convex polytopes

Let P be a convex polytope in R, 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 algorithm, where the role of vertices is...

Fine Approximation of Convex Bodies by Polytopes

We prove that for every convex body K with the center of mass at the origin and every ε ∈ ( 0, 12 ) , there exists a convex polytope P with at most eO(d)ε− d−1 2 vertices such that (1− ε)K ⊂ P ⊂ K.

Convex polytopes and linear algebra

This paper defines, for each convex polytope ∆, a family Hw∆ of vector spaces. The definition uses a combination of linear algebra and combinatorics. When what is called exact calculation holds, the dimension hw∆ of Hw∆ is a linear function of the flag vector f∆. It is expected that the Hw∆ are examples, for toric varieties, of the new topological invariants introduced by the author in Local-gl...

Feasible Region Approximation Using Convex Polytopes

A new technique for polytope approximation of the feasible region for a design is presented. This method is computationally less expensive than the simplicial approximation method [1]. Results on several circuits are presented, and it is shown that the quality of the polytope approximation is substantially better than an ellipsoidal approximation.