Geodesic
Carathéodory’s Theorem with Linear Constraints
Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 189-191

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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). 
Cook, W. D. Carathéodory’s Theorem with Linear Constraints. Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 189-191. doi: 10.4153/CMB-1974-038-0
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     author = {Cook, W. D.},
     title = {Carath\'eodory{\textquoteright}s {Theorem} with {Linear} {Constraints}},
     journal = {Canadian mathematical bulletin},
     pages = {189--191},
     year = {1974},
     volume = {17},
     number = {2},
     doi = {10.4153/CMB-1974-038-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-038-0/}
}
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