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Kinematic measures for sets of support figures

Part of: Global differential geometry General convexity

Published online by Cambridge University Press:  26 February 2010

William J. Firey
William J. Firey
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, U.S.A.
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Extract

1. W. Blaschke's kinematic formula in the integral geometry of Euclidean n-dimensional space gives a weighted measure to the set of positions in which a mobile figure K1 overlaps a fixed figure K0. In the simplest case, K0 and K1 are compact convex sets and all positions are equally weighted; we give this in more detail. Let Wq denote the q-th Quermassintegral of K1: Steiner's formula for the volume V of the vector sum K1 + λB of K1 and a ball of radius λ defines these set functions by the equation

see [4; p. 214]. Blaschke's formula [4; p. 243] gives

as the measure, to within a normalization, of overlapping positions of K1 relative to K0.

MSC classification

Secondary: 52A40: Inequalities and extremum problems 53C65: Integral geometry
Type
Research Article
Information
Mathematika , Volume 21 , Issue 2 , December 1974 , pp. 270 - 281
Copyright
Copyright © University College London 1974

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References

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