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 Santaló [4] with other methods.
In the appendix, the section formula is applied to motion invariant random sets.
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Rother, W., Zähle, M. Absolute curvature measures, II. Geom Dedicata 41, 229–240 (1992). https://doi.org/10.1007/BF00182423
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DOI: https://doi.org/10.1007/BF00182423