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Recall that any sufficiently nice compact centrally symmetric convex body in $B \subset \mathbb{R}^3$ gives rise to a Banach norm on $\mathbb{R}^3$ which has $B$ as its unit ball.

Is there a nice characterization of those norms arising from the centrally symmetric Platonic solids?

I'm mostly interested in packing properties, covering estimates, doubling constants.

Edit: One dimension lower you'd have things like: One can tile the plane with fixed radius balls only if your norm arises from a square or hexagon. This should imply that coverings of sets by balls of norms not associated with the square and hexagon would necessarily be inefficient.

Edit: The norms arising from the cube and octohedron must have some known characterization, since they are $\ell^\infty(\mathbb{R}^3)$ and $\ell^0(\mathbb{R}^3)$, which have been studied. The only other centrally symmetric platonic solids are the dodecahedron and icosahedron, so this question is quintessentially about them.

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  • $\begingroup$ An obvious necessary condition is that the isometries fixing the origin must the same as the isometries of the solid. $\endgroup$
    – Deane Yang
    Jan 28, 2014 at 1:06

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