### Abstract

The authors give an O(n
^{7/log}
sup 2/n) randomised algorithm to approximate the volume of a convex body, and an O(n
^{6}
log n) algorithm to sample a point from the uniform distribution over a convex body. For convex polytopes the algorithm runs in O(n
^{7/log}
sup 4/n) steps. Several tools are developed that may be interesting on their own. They extend results of Sinclair-Jerrum (1988) and the authors (1990) on the mixing rate of Markov chains from finite to arbitrary Markov chains. They describe an algorithm to integrate a function with respect to the stationary distribution of a general Markov chain. They also analyze the mixing rate of various random walks on convex bodies, in particular the random walk with steps from the uniform distribution over a unit ball. In several previous positive and negative results, the problem of computing the diameter of a convex body behaved similarly as the volume problem. In contrast to this, they show that there is no polynomial randomized algorithm to compute the diameter within a factor of n
^{1/4}
.

Original language | English |
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Title of host publication | Proceedings - 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 |

Publisher | IEEE Computer Society |

Pages | 482-491 |

Number of pages | 10 |

ISBN (Electronic) | 0818629002 |

DOIs | |

Publication status | Published - Jan 1 1992 |

Event | 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 - Pittsburgh, United States Duration: Oct 24 1992 → Oct 27 1992 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 1992-October |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 |
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Country | United States |

City | Pittsburgh |

Period | 10/24/92 → 10/27/92 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings - 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992*(pp. 482-491). [267803] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 1992-October). IEEE Computer Society. https://doi.org/10.1109/SFCS.1992.267803