  Mirkovi\'c-Vilonen polytopes encode in a geometrical way the numerical data
present in the Kashiwara crystal $B(\infty)$ of a semisimple group $G$. We
retrieve these polytopes from the coproduct of the Hopf algebra $\mathscr O(N)$
of regular functions on a maximal unipotent subgroup $N$ of $G$. We bring
attention to a remarkable behavior that the classical bases (dual canonical,
dual semicanonical, Mirkovi\'c-Vilonen) of $\mathscr O(N)$ manifest with
respect to the extremal points of these polytopes, which extends the crystal
operations. This study leans on a notion of stability for graded bialgebras.
