The existence of $x'_i$ is referred to as the cheaper point assumption. The figure demonstrates what can go wrong with the duality between expenditure minimization and utility maximization when the cheaper point assumption does not hold.
The consumption set is $\mathbb{R^2_{+}}$ in which the open triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ has been removed. Prices and wealth are such that the budget set is the lower 45 degree line. The red lines are indifference curves. The indicated consumption bundle is expenditure minimizing on its 'no worse than' set, but it is not preference maximal on the budget set.
