The pattern of this chapter closely follows that of the preceding one. The definition of area that is explored in this chapter was investigated intensively by Busemann in a series of important papers [66, 68,71]. The idea, however, goes back to Bouligand and Choquet [53]. Going further back, we will also show in §7.3 that the definition agrees with the (d − 1)-dimensional Hausdorff measure arising from the metric on (X, B). Since the solution to the isoperimetric problem IB that this definition yields is the dual of the intersection body of B, recent work on intersection bodies is relevant. We refer in particular to Lutwak [337, 340], Ball [14], Milman and Pajor [382], Gardner [168, 169] and Zhang [541, 542, 544]. All the material on intersection bodies used in this chapter can be found in Gardner [172].

The convexity of the area function σ

We begin by recalling Busemann's definition; see Example 5.1.4(i) and Equations (5.7) and (5.8). Given a Minkowski space (X, B) Haar measure in each hyperplane in X is normalized by prescribing the (d − 1)-dimensional measure μB of the cross-section (B ⋂ f⊥) of the unit ball by a hyperplane f⊥ to be

Furthermore, if λ is an auxiliary Lebesgue measure then the function σB(f), defined first for unit vector as the ratio of Minkowski measure to Lebesgue measure in hyperplanes parallel to and then made positively homogeneous in f, is given by

Unless specific reference to B is needed the subscript will be omitted.