Golab’s theorem

Theorem. Let $D$ be the unit disc of a Minkowski plane and let$\mathrm{\ell }(\partial D)$ denote the Minkowski length (http://planetmath.org/LengthOfCurveInAMetricSpace) of the boundary of $D$. Then$6\le \mathrm{\ell }(\partial D)\le 8$. The lower bound is attained if and onlyif $D$ is linearly equivalent to a regular hexagon. The upper boundis attained if and only if $D$ is a parallelogram.

Note that 1/2 the perimeter of the unit disc is a constantbetween 3 and 4. The special case of the 2-norm yields a constant,which is known as $\pi$. So Golab’s theorem is that ”pi” for aMinkowski plane is always between 3 and 4.