Abstract
Mixed curvature measures for sets of positive reach are introduced and a translative version of the principal kinematic formula from integral geometry is proved. This is an extension of a known result from convex geometry. Integral representations of the mixed curvature measures in various particular cases of dimension two and three are derived.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Federer, H.: Curvature measures,Trans. Amer. Math. Soc. 93 (1959), 418–491.
Federer, H.:Geometric Measure Theory, Springer Verlag, Berlin, 1969.
Goodey, P. and Weil, W.: Translative integral formulae for convex bodies,Aequationes Math. 34 (1987), 64–77.
McMullen, P.: Monotone translation invariant valuations on convex bodies,Arch. Math. 55 (1990), 595–598.
Rataj, J.: Estimation of oriented direction measure of a planar body,Adv. Appl. Probab. 28 (1996), to appear.
Rother, W. and Zähle, M.: A short proof of a principal kinematic formula and extensions,Trans. Amer. Math. Soc. 321 (1990), 547–558.
Schneider, R.:Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.
Schneider, R. and Weil, W.: Translative and kinematic integral formulae for curvature measures,Math. Nachr. 129 (1986), 76–80.
Schneider, R. and Weil, W.:Integralgeometrie, Teubner, Stuttgart, 1992.
Scholtes, S.: Convex bodies in two dimensions,Mathematika 39 (1992), 267–273.
Weil, W.: Iterations of translative integral formulae and non-isotropic Poisson processes of particles.Math. Z. 205 (1990), 531–549.
Zähle, M.: Random processes of Hausdorff rectifiable closed sets,Math. Nachr. 108 (1982), 49–72.
Zähle, M.: Integral and current representation of Federer's curvature measures,Arch. Math. 46 (1986), 557–567.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rataj, J., Zähle, M. Mixed curvature measures for sets of positive reach and a translative integral formula. Geom Dedicata 57, 259–283 (1995). https://doi.org/10.1007/BF01263484
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01263484