On the geometry of two- and three-dimensional Minkowski spaces

A class of centrally-symmetric convex 12-topes (12-hedrons) in $\mathbb R^3$ is described, such that for an arbitrary prescribed norm ${\|\cdot\|}$ on $\mathbb R^3$ each polyhedron in the class can be inscribed in (circumscribed about) the ${\|\cdot\|}$-ball via an affine transformation, and this can be done with large degree of freedom. It is also proved that the Banach–Mazur distance between any two two-dimensional real normed spaces does not exceed $\ln(6-3\sqrt2)$.