The problem to compute a V-polytope which is close to a given H-polytope is addressed. Both polytopes are not required to be combinatorially equivalent. This raises the question whether  approximate vertex enumeration  is easier to realize than exact vertex enumeration. Beyond direct potential applications, approximate vertex enumeration could serve as a model to understand numerical problems with floating point implementations of exact methods. An approximate variant of Motzkin's double description method  is developed. Under certain  conditions we are able to control the approximation error and to prove correctness of the algorithm for arbitrary polytopes. For dimension 2 and 3 these conditions can be omitted, which allows an easy implementation of the method.  It remains open if the conditions are required for dimension larger than 3.
The problem to compute a V-polytope which is close to a given H-polytope is addressed. Both polytopes are not required to be combinatorially equivalent. This raises the question whether this problem, called approximate vertex enumeration , is easier to realize than exact vertex enumeration. An approximate variant of Motzkin's double description method (DDM) is developed. Under certain strong conditions we are able to control the approximation error and to prove correctness of the algorithm for arbitrary polytopes. For dimension 2 and 3 these conditions can be omitted, which allows an easy implementation of the method. We provide an algorithm for the approximate vertex enumeration problem which is correct in dimension 2 and 3 even if imprecise arithmetic is used. It remains open if the latter results can be extended to a dimension larger than 3.
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this problem, called
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,
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Beyond direct potential applications, approximate vertex enumeration could serve as a model to understand numerical problems with floating point implementations of exact methods.
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(DDM)
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strong
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We provide an algorithm for the approximate vertex enumeration problem which is correct in dimension 2 and 3 even if imprecise arithmetic is used.
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conditions are required for
latter results can be extended to a