1. Introduction

The study of volumes and areas on normed and Finsler spaces is a relatively new field that comprises and unifies large domains of convexity, geometric tomography, and integral geometry. It opens many classical unsolved problems in these fields to powerful techniques in global differential geometry, and suggests new challenging problems that are delightfully geometric and simple to state.

The theory starts with a simple question: How does one measure volume on a finite-dimensional normed space? At first sight, this question may seem either unmotivated or trivial: normed spaces are metric spaces and we can measure volume using the Hausdorff measure, period. However, if one starts asking simple, naive questions one discovers the depth of the problem. Even if one is unwilling to consider that definitions of volume other than the Hausdorff measure are not only possible but may even be better, one is faced with questions such as these: What is the (n−1)-dimensional Hausdorff measure of the unit sphere of an n-dimensional normed space? Do flat regions minimize area? For what normed spaces are metric balls also the solutions of the isoperimetric problem? These questions, first posed in convex-geometric language by Busemann and Petty [1956], are still open, at least in their full generality. However, one should not assume too quickly that the subject is impossible. Some beautiful results and striking connections have been found. For example, the fact that the (n−1)-Hausdorff measure in a normed space determines the norm is equivalent to the fact that the areas of the central sections determine a convex body that is symmetric with respect to the origin.