On the Intersection of Two Particular Convex Sets

In this note, necessary and sufficient conditions are given for the intersection of the m-1 simplex co{ξ1, . . . , ξm} of m affinely independent vectors ξ1, . . . , ξmof ℝnand the negative orthant ℝn-to be empty, i.e., co{ξ1, . . . , ξm} ∩ ℝn-= ∅, where m ≤ n. It is also shown that the special case m=2 can be checked easily. These results suggest that the above-mentioned emptiness can be checked recursively. Some numerical examples are given to illustrate the results. Potential applications of these results include the compatible multicommodity flow problems and satisficing solution problems.