An Euler-type Version of the Local Steiner Formula for Convex Bodies 1 Theorem 1. Let K 2 K N Be a Convex Body. Then

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 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 (1) for all integrable functions f : R n ! R; here denotes the Lebesgue measure on R n. Equation (1), which is a consequence and a slight generalization of Theorem 4.2.1 in Schneider 9], is called the local Steiner formula. Our main result is the following: