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Computing the volume is difficult. (English) Zbl 0628.68041
Assuming the black box model of a convex set in d-dimensional Euclidean space, algorithms for computing the volume or the width of convex sets are analyzed. Negative results are proved: For every polynomial time algorithm which gives an upper bound $$\overline{vol}$$ and a lower bound $$\underline{vol}$$ for the volume of a convex set in d-dimensional Euclidean space, there exists a convex set such that the ratio $$\overline{vol}/\underline{vol}$$ is greater than $$(cd/\log d)^ d$$. Similarly, for polynomial time algorithms calculating the width of a convex set (bounds $$\bar w$$ and $$\underline w$$), there exists a convex set such that $$\bar w/\underline w>(d/(c \log d))^{1/2}$$.
Reviewer: R.Klette

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 52A37 Other problems of combinatorial convexity 52A40 Inequalities and extremum problems involving convexity in convex geometry
##### Keywords:
volume computation; time complexity; width of convex sets
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##### References:
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