On the intersection of two particular convex sets

In this note, necessary and sufficient conditions are given for the intersection of them?1 simplex co {ξ¹,...,ξ ^m } ofm affinely independent vectors ξ¹,...,ξ ^m of ? ⁿ and the negative orthant ? _? ⁿ to be empty, i.e., $$co\{ \xi ^1 ,...,\xi ^m \} \cap \mathbb{R}_ - ^n = \emptyset ,$$ wherem≤n. It is also shown that the special casem=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.