Absolute curvature measures, II

Abstract

The absolute curvature measures for sets of positive reach in R d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santal6 [4] with other methods.In the appendix, the section formula is applied to motion invariant random sets. EQUIVALENT DEFINITIONSThe present paper is a continuation of [7], which also contains motivations and references to the related literature. We first recall some notions introduced there. At the same time we correct an error in formulas (3)-(6) of [7], where certain projection Jacobians have to be inserted. In the sequel, X denotes an arbitrary subset of R d of positive reach (in the sense of Federer [1]) and B a bounded Borel set in R d x S d-1 (or in Rd). Note that convex bodies and compact C2-submanifolds are special cases for X. In [7] the kth absolute curvature measure c~bs(x, B), k = 0 ..... d-1, is introduced by means of the natural invariant measure of the set of all (d-1 -k)-planes locally colliding with X inside B. Thereby affine and linear subspaces of R d are considered as dements of sufficiently large Euclidean spaces via representation by multivectors. Let nor X be the unit normal bundle of X and G(X, d, k) its kth Grassmann bundle (denoted in [7] by G(X, k)), i.e. G(X, d, k)= {(x, n, V): (x, n)~nor X, V~G,(d-1, k)} where G,(d-1, k) is the Grassmann submanifold of G(d, k) of those kdimensional linear subspaces of R d which are orthogonal to the unit vector n ES d-1. Let v" be the normalized rotation invariant measure on d-l,k G,(d-l,k). 17 v is the orthogonal projection onto the subspace V The function f(x, n, V) = (IIv~ x, V) maps G(X, d, k) onto the set sO(X, d, k) of parametrized k-dimensional affine subspaces (k-planes) locally colliding with X. Let P be the coordinate Geometriae Dedicata 41: 229-240, 1992.

Absolute curvature measures for unions of sets with positive reach

“…Then nor X is H' l~l -measurable and locally (d -l)-integrable and, at H d~lalmost all (x. n) e nor X, the (generalized) principal directions a,(x,ri) e S d~l and (generalized) principal curvatures K,(X, ri) e (-00, 00] (/ = 1,..., d -1) are denned so that the (H''~\d -l)-approximate tangent cone Tan''" 1 (nor X, (x, ri)) is a {d -1 )-dimensional linear subspace of K 2 '' spanned by the vectors Remark. The analogous result for sets with positive reach was stated and used in [10]. We present a proof here for the sake of completeness.…”

“…(Note that, in particular, any set from the convex ring is a UPR set.) The definition presented extends that of Zahle [10,13] by considering the index function of UPR sets. It is shown that the Crofton intersection formula for absolute curvature measures is still valid for UpR-sets; this is achieved by generalizing a translative intersection formula for sets with positive reach [6].The definition presented makes it possible to compare the two approaches appearing in the literature and mentioned above to define non-negative variants of intrinsic volumes and curvature measures, namely, that of Matheron and Schneider with the integral-geometric one (Santalo, Baddeley, and Zahle), since the system of locally finite unions of convex bodies is covered by both of them.…”

“…(Note that, in particular, any set from the convex ring is a UPR set.) The definition presented extends that of Zahle [10,13] by considering the index function of UPR sets. It is shown that the Crofton intersection formula for absolute curvature measures is still valid for UpR-sets; this is achieved by generalizing a translative intersection formula for sets with positive reach [6].…”

Absolute continuity for curvature measures of convex sets II

“…Let us fix a convex body K ∈ K d and some [51], [18], [54], [35], [44] have introduced naturally defined measures. For convex bodies, however, all these measures are essentially equivalent.…”

“…2. A parametrization of this rectifiable set is provided in [54] and [35]. We say that K is supported from outside by an orthogonal spherical cylinder at E ∈ A (K, d, r) if there is some R > 0 and some…”