Steiner Formula for Convex Bodies

This dissertation provides a detailed exposition of the Steiner formula and the associated theory of curvature measures for general convex bodies in Euclidean space. The core challenge addressed is the extension of differential geometric concepts like curvature to non-smooth convex sets, such as polytopes. To this end, we first establish the necessary analytic groundwork. We treat the space of convex bodies as a complete metric space under the Hausdorff metric and as a measurable space, proving essential continuity and compactness properties. Central to this analysis is the study of the metric projection and its properties. Equipped with this foundation, we undertake a systematic development of the Steiner formula. The proof proceeds by first establishing the formula for convex polytopes, leveraging results on face structure and support functions. The general case for arbitrary convex bodies is then obtained via approximation by polytopes and the continuity of the volume functional. This process naturally leads to the definition of curvature measures and surface area measures as the coefficients in the Steiner polynomial, thereby assigning a rigorous notion of curvature to all convex bodies, regardless of smoothness. The results presented form a complete bridge from the basic metric properties of convex sets to the core geometric identities of the Brunn–Minkowski theory, with the Steiner formula and curvature measures as the pivotal link.