The volume of the convex hull of anym points of ann-dimensional ball with volumeV is at mostV·m/2n. This implies that no polynomial time algorithm can compute the volume of a convex set (given by an oracle) with less than exponential relative error. A lower bound on the complexity of computing width can also be deduced.
M. Grötschel, L. Lovász, and A. Schrijver, Geometric methods in combinatorial optimization, inProgress in Combinatorial Optimization, Vol. 1 (W. R. Pulleyblank, ed.), 167–183, Academic Press, New York, 1984.
L. Lovász,An Algorithmic Theory of Numbers, Graphs, and Convexity, AMS-SIAM Regional Conference Series, to appear.
Dedicated to my teacher Kõváry Károly
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Elekes, G. A geometric inequality and the complexity of computing volume. Discrete Comput Geom 1, 289–292 (1986). https://doi.org/10.1007/BF02187701
- Polynomial Time
- Convex Hull
- Convex Body
- Polynomial Time Algorithm
- Discrete Comput Geom