Abstract
It is shown that ifF is ann-dimensional exposed face of ad-dimensional convex body and iff is a linear functional whose maximum value on the body is attained over the whole ofF thenn+1 paths can be found in the one-skeleton of the body leading toF and disjoint except at their end-points. Further, such paths may be found having the property that along them, the value off strictly increases. It is further shown that unlessn=0 it may be impossible to findn+2 such paths.
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