A short proof of principal kinematic formula and extensions

Authors:
W. Rother and M. Zähle

Journal:
Trans. Amer. Math. Soc. **321** (1990), 547-558

MSC:
Primary 53C65; Secondary 49Q15, 58A25

MathSciNet review:
987167

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Federer's extension of the classical principal kinematic formula of integral geometry to sets with positive reach is proved in a direct way by means of generalised unit normal bundles, associated currents, and the coarea theorem. This enables us to extend the relation to more general sets. At the same time we get a short proof for the well-known variants from convex geometry and differential geometry.

**[1]**Shiing-shen Chern,*On the kinematic formula in integral geometry*, J. Math. Mech.**16**(1966), 101–118. MR**0198406****[2]**Herbert Federer,*Curvature measures*, Trans. Amer. Math. Soc.**93**(1959), 418–491. MR**0110078**, 10.1090/S0002-9947-1959-0110078-1**[3]**-,*Geometric measure theory*, Springer, Berlin, 1969.**[4]**Joseph H. G. Fu,*Kinematic formulas in integral geometry*, Indiana Univ. Math. J.**39**(1990), no. 4, 1115–1154. MR**1087187**, 10.1512/iumj.1990.39.39052**[5]**Luis A. Santaló,*Integral geometry and geometric probability*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR**0433364****[6]**M. Zähle,*Curvature measures and random sets. I*, Math. Nachr.**119**(1984), 327–339. MR**774200**, 10.1002/mana.19841190129**[7]**M. Zähle,*Integral and current representation of Federer’s curvature measures*, Arch. Math. (Basel)**46**(1986), no. 6, 557–567. MR**849863**, 10.1007/BF01195026**[8]**M. Zähle,*Curvatures and currents for unions of sets with positive reach*, Geom. Dedicata**23**(1987), no. 2, 155–171. MR**892398**, 10.1007/BF00181273**[9]**-,*Normal cycles and second order rectifiable sets*(submitted).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53C65,
49Q15,
58A25

Retrieve articles in all journals with MSC: 53C65, 49Q15, 58A25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0987167-X

Article copyright:
© Copyright 1990
American Mathematical Society