### Abstract

For a finite set of distinct points S = {p_{i} ε l} in ℝ^{d} there exists Î ⊆ I such that all points in Ŝ = {p_{i} i ε Î} are extreme points and conv(Ŝ) = conv(S). Since a point p_{k} ε S is extreme if and only if the inequality p^{T}_{k} y ≤ 1 is necessary with respect to the representation {y ε ℝ^{d}{pipe} p^{T}_{i} y≤ 1, i ε I} of the polar dual S^{Δ} of S, Ŝ can be found by classifying the inequalities in the representation as necessary or redundant. Thus, the problem of finding Ŝ is polynomial. This paper shows the advantage of using probabilistic hit-and-run algorithms applied to the polar dual for the quick identification of points in Ŝ and shows how, in an application to a certain cluster analysis problem, it can be used to also identify points in S/Ŝ. Further, it shows that the hit-and-run variant known as Stand-and-Hit provides approximations for the exterior solid angles at the extreme points of S.

Original language | English (US) |
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Pages (from-to) | 131-142 |

Number of pages | 12 |

Journal | Journal of Interdisciplinary Mathematics |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - 2007 |

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### Keywords

- Convex hull
- Extreme points
- Linear inequalities
- Probabilistic method
- Redundancy
- Solid angles

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Interdisciplinary Mathematics*,

*10*(1), 131-142. https://doi.org/10.1080/09720502.2007.10700478