The Steiner Formula for Convex Subsets

This chapter is devoted to the computation of the volume of the parallel body of a convex body Kat distance ε (see the definition below). It appears that the convexity of Kimplies that this volume is polynomial in ε, the coefficients (Φ _k (K),0≤k ≤N) depending on the geometry of K[77]. Up to a constant, these coefficients (called the Quermassintegrale of Minkowski) are the valuations, which appear in Definition 23 and Theorem 28 of Hadwiger. Moreover, these coefficients can be easily evaluated when the boundary of Kis smooth: up to a constant depending on N, they are the integral of the k^th-mean curvatures of the boundary ∂K of K. That is why they are good candidates to generalize the curvatures of a smooth hypersurface: they can be defined for any convex subset, even if its boundary is not of class C². We shall say that the sequence (Φ _k (K) defines the k^th-mean curvatures of of K(and by extension if there is no possible confusion, the k^th-mean curvatures of of ∂K). Of course, the explicit evaluation of these curvatures cannot be done by differentiations of a parametrization of the boundary, because of the lack of differentiability. We shall directly evaluate them for convex polyhedra. All these techniques will be generalized in the next chapters to objects which are not convex, but which have geometrical properties close to those of convex bodies.