The PBW filtration and convex polytopes in type $\mathtt{B}$

We study the PBW filtration on irreducible finite–dimensional representations for the Lie algebra of type ${\mathtt{B}}_{\mathtt{n}}$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite–dimensional representations for ${\mathtt{B}}_{\mathtt{3}}$, that there exists a normal polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence we obtain several classes of favourable modules and graded combinatorial character formulas.