How Come RIP Guarantees Unique Restoration of the Sparse Solution by $ {L}_{1} $ Minimization?
Compressed sensing theory (see for example Candès &amp; Wakin, 2008) states that, given you have enough measurements $y$, the shown $\ell_1$ minimisation problem recovers $x$ exactly with very high probability. This means that your correct solution must have the smallest possible $\ell_1$ norm.

This is what the figure shows; the "$\ell_1$ ball" intersects the plane given by $\Phi x' = y$ at the point $x'$ with the smallest $\ell_1$ norm. There are, as you mention, other sparse solutions - i.e. the ones where the plane intersects the other two axes. As we can imagine from the figure, these points however have larger $\ell_1$ norm and are therefore not the correct solution.