The case of two-dimensional normed spaces is, in some respects, quite special. In other ways, it affords a good introduction to higher-dimensional geometry. This was evident in the last chapter. The results in the earlier sections dealing with circumscribed parallelotopes and inscribed and circumscribed ellipsoids are no different in dimension 2 than in other dimensions. On the other hand, in §3.4 the characterizations of Euclidean space via norm 1 projections and via the symmetry of orthogonality are only valid in dimension d ≤ 3. When d = 2 these conditions lead to an interesting class of convex bodies.

From the point of view of our main topic – the normalization of Haar measure in all the subspaces of a Minkowski space – the particularity of dimension 2 comes from the fact that the only non-trivial subspaces are lines and there the normalization is already specified by the metric, i.e. by the unit ball. Thus, except for the choice of the single normalizing factor for Haar measure in the whole space, which affects the numerical value to be assigned to certain important concepts but not any of the theoretical framework, the geometry is entirely determined by the unit ball.

That this is not so in higher dimensions, that one has a certain amount of freedom in the choice of the isoperimetrix (the convex body of minimal surface area for a given volume), forms the content of Chapter 5. It is important, therefore, to have the unambiguous solution in two dimensions clearly determined first. Whatever solution is offered in higher dimensions, one of the constraints must be that it specializes to the right object when d = 2.