An Euler-Type Version of the Local Steiner Formula for Convex Bodies

Abstract

The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann [3] and Hug [6], who use much less elementary techniques than the methods of this paper. In fact, Hann [4] asked for a simpler proof of these results ([4], Problem 2 on p. 900). We remark that our formula can be considered as a Euclidean analogue of a spherical result proved in [2], p. 46, and that our method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensional Minkowski spaces, see Hann [5], p. 363. For further information about the notions we use below we refer to Schneider’s book [9]. Let K be the set of all convex bodies in Euclidean space R, i.e. the set of all compact, convex, non-empty subsets of R. Let Sn−1 be the unit sphere. For K ∈ K, let Nor K be the set of all support elements ofK, i.e. the pairs (x, u) ∈ Rn×Sn−1 such that x is a boundary point of K and u is an outer unit normal vector of K at the point x. The support measures (or generalized curvature measures) of K, denoted by Θ0(K, ·), . . . ,Θn−1(K, ·), are the unique Borel measures on R × Sn−1 that are concentrated on Nor K and satisfy ∫