Elementary, constructive proofs of the theorems of Farkas, Minkowski and Weyl. (English) Zbl 0718.52005
Economic decision-making: games, econometrics and optimisation, Contrib. in Honour of J. H. Drèze, 427-432 (1990).
[For the entire collection see Zbl 0707.00031.]
We recall that for points \(x_ 1,...,x_ n\) in \({\mathbb{R}}^ d\), \[ pos(x_ 1,...,x_ n)=\{\sum^{n}_{i=1}\lambda_ ix_ i| \lambda_ i\geq 0\text{ for } i=1,...,n\} \] is their positive hull; furthermore, \(pos(x_ 1,...,x_ n)\) is a closed convex cone (Farkas’s theorem) and the intersection of a finite number of half-spaces whose bounding hyperplanes pass through the origin (Weyl’s theorem).
The author presents a nice detailed analysis with elementary and constructive proofs of these results along with that of Minkowski for convex polyhedral cones.
We recall that for points \(x_ 1,...,x_ n\) in \({\mathbb{R}}^ d\), \[ pos(x_ 1,...,x_ n)=\{\sum^{n}_{i=1}\lambda_ ix_ i| \lambda_ i\geq 0\text{ for } i=1,...,n\} \] is their positive hull; furthermore, \(pos(x_ 1,...,x_ n)\) is a closed convex cone (Farkas’s theorem) and the intersection of a finite number of half-spaces whose bounding hyperplanes pass through the origin (Weyl’s theorem).
The author presents a nice detailed analysis with elementary and constructive proofs of these results along with that of Minkowski for convex polyhedral cones.
Reviewer: T.Bisztriczky (Calgary)
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52B11 | \(n\)-dimensional polytopes |
