The theory of mixed volumes is a powerful tool for treating some questions on closed convexhypersurfaces from the point of view of differential geometry, but in a general form withoutdifferentiability assumptions. Under smoothness assumptions, the results we have in mind concern thedetermination of closed convex hypersurfaces from curvature functions, such as Gauss curvature, meancurvature and their generalizations. Here ‘determination’ comprises questions ofexistence, uniqueness and stability. Without differentiability assumptions, the usual curvaturefunctions, namely the elementary symmetric functions of the principal curvatures on the boundary ofa convex body or of the principal radii of curvature on the spherical image, have to be replaced bycurvature measures and area measures, respectively. The area measures are particularly accessible tothe Brunn–Minkowski theory. In Section 8.1 we treat uniqueness theorems for these. Section8.2 is devoted to Minkowski's existence theorem for convex bodies with given surface areameasure (area measure of order n – 1) and Section 8.3 deals with areameasures of order one, where the existence problem is known as the Christoffel problem. Theintermediate cases, area measures of orders strictly between 1 and n – 1,are briefly considered in Section 8.4. The final section is devoted to corresponding stabilityestimates and to a few uniqueness results for curvature measures.

Uniqueness results

We start with the uniqueness assertion for the determination of a convex body by its surface areameasure. Although this result will be improved and generalized by later theorems, we give itsformulation and proof separately, to show in a basic example the close connection with results onmixed volumes.