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Carathéodory’s Theorem with Linear Constraints

Published online by Cambridge University Press:  20 November 2018

W. D. Cook
W. D. Cook*
Affiliation:
Faculty of Administrative Studies, York University, Toronto, Ontario
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Carathéodory has shown that if x1, x2,…, xm, (m finite) are points of Rn and if some

then ∃μ∈Ω with at most n+1 nonzero components and for which (See [5]). The authors of [2] have extended this result to include the case where m= +∞. In theorems 1 and 2 below we establish somewhat similar results for the case in which Ω is further restricted by a finite system of linear inequalities (or equalities).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 189 - 191
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Charnes, A., Cooper, W. W. and Kortanek, K., “Duality in semi-infinite programmes and some works of Haar and Carathéodory”, Man. Sci. 9 (1963), 209-228.Google Scholar
2. Cook, W. D. and Webster, R. J., “Carathéodorfs theorem”, Canadian Math. Bull. 15 (1972), 888.Google Scholar
3. Haar, A., “Uber Linear Ungleichungen”, Acta. Math. 2 (1924), 1-14.Google Scholar
4. Hadley, G., Linear programming, Addison-Wesley, Reading, Mass. (1962).Google Scholar
5. Reay, J. R., Generalizations of a theorem of Carathéodory, Memoirs Amer. Math. Soc. 54 (1965).Google Scholar