Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-19T11:18:41.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Kinematic measures for sets of support figures

Part of: General convexity Global differential geometry

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Carathéodory, C.. Mass und Integral und ihre Algebrasierung (Basel and Stuttgart, 1956).CrossRefGoogle Scholar
2.Fenchel, W. and Jessen, B.. “Mengenfunktionen und konvexe Körper”, Del Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd., 16 (1938), 3.Google Scholar
3.Firey, W. J.. “An integral-geometric meaning for lower order area functions of convex bodies”, Mathematika, 19 (1972), 205–212.CrossRefGoogle Scholar
4.Hadwiger, H.. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Berlin, Göttingen, Heidelberg, 1957).CrossRefGoogle Scholar
5.McMullen, P.. “A dice probability problem”, Mathematika, 21 (1974), 193–198.CrossRefGoogle Scholar
6.Nachbin, L.. The Haar Integral (Princeton, 1965).Google Scholar
7.Pogorelov, A. V.. The extrinsic geometry of convex surfaces (Moscow, 1969) (Russian). English translation under the same title (Providence, 1973).Google Scholar
8.Santalo, L. A.. Integral Geometry (Paris, 1953).Google Scholar