Abstract
We show that the natural “convolution” on the space of smooth, even, translation-invariant convex valuations on a euclidean space V, obtained by intertwining the product and the duality transform of S. Alesker J. Differential Geom. 63: 63–95, 2003; Geom.Funct. Anal. 14:1–26, 2004 may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valuations as well. Based on this technical result we give an application to integral geometry, generalizing Hadwiger’s additive kinematic formula for SO(V) Convex Geometry, North Holland, 1993 to general compact groups \(G \subset O(V)\) acting transitively on the sphere: it turns out that these formulas are in a natural sense dual to the usual (intersection) kinematic formulas.
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References
Alesker S. (2001). Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Func. Anal. 11: 244–272
Alesker S. (2003). Hard Lefschetz theorem for valuations, complex integral geometry and unitarily invariant valuations. J. Differential Geom. 63: 63–95
Alesker, S.: Hard Lefschetz theorem for valuations and related questions of integral geometry. In: Milman, V.D. (ed.) Geometric aspects of functional analysis, pp. 9–20, LNM 1850. Springer, Berlin,Heidelberg Newyork (2004)
Alesker S. (2004). The multiplicative structure on polynomial valuations. Geom. Funct. Anal. 14: 1–26
Alesker, S.: Theory of valuations on manifolds I. Linear spaces. To appear in Israel J. Math.
Alesker S. (2006). Theory of valuations on manifolds II. Adv. Math. 207: 420–454
Alesker, S.: Theory of valuations on manifolds IV. New properties of the multiplicative structure. To appear in GAFA seminar notes
Alesker, S.: Valuations on manifolds: a survey. Preprint 2006.
Alesker, S., Fu, J.H.G.: Theory of valuations on manifolds III. Multiplicative structure in the general case. To appear in Trans. Amer. Math. Soc.
Bernig, A.: Valuations with Crofton formula and Finsler geometry. To appear in Adv. Math.
Bernig, A., Bröcker, L.: Valuations on manifolds and Rumin cohomology. To appear in J. Differential Geom.
Federer H. (1969). Geometric Measure Theory. Springer-Verlag, New York
Fu J.H.G. (2006). Structure of the unitary valuation algebra. J. Differential Geom. 72: 509–533
Klain D. (2000). Even valuations on convex bodies. Trans. Amer. Math. Soc. 352: 71–93
Klain D. and Rota G.-C. (1997). Introduction to Geometric Probability. Lezione Lincee Cambridge University Press, Cambridge
McMullen P. (1977). Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. 35: 113–135
Schneider R and Wieacker J.A. (1993). Integral geometry. In: Gruber, P.M. and Wills, J.M. (eds) Handbook of Convex Geometry vol B, pp. Amsterdam, North Holland
Whitney H. (1957). Geometric integration theory. Princeton University Press, princeton
Zähle M. (1986). Integral and current representation of Federer’s curvature measures. Arch. Math. 46: 557–567
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Bernig, A., Fu, J.H.G. Convolution of convex valuations. Geom Dedicata 123, 153–169 (2006). https://doi.org/10.1007/s10711-006-9115-7
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DOI: https://doi.org/10.1007/s10711-006-9115-7