Is it possible to have 2 different points (non-degenerate) in $n$-dimensions for any value of $n$ where $n>1$, share the same cost and both be visited by the simplex method?
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At each iteration the simplex method looks for an improving feasible direction. If there is one, then the next basic feasible solution has a lower cost than the current one (except in the case of degeneracy, which you've already ruled out). Since the costs of the basic feasible solutions are thus strictly decreasing (again, except for degeneracy) there cannot be two nonoptimal points with the same cost that are visited by the simplex method.
If there isn't an improving feasible direction, then the simplex method stops, because it has found an optimal solution. So it wouldn't visit two optimal points with the same cost, either.
Since two such points would both have to be optimal or both be nonoptimal, it appears they cannot exist.