Convolution of convex valuations

Details Convolution of convex valuations Convolution of convex valuations

2006-07-19

arxiv.org/abs/math/0607449v2

Geom. Dedicata 123 (2006), 153-169

Mathematics

Differential Geometry

18 pages; Thm. 1.4. added; references updated; other minor changes; to appear in Geom. Dedicata

Scientific paper

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, 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 $SO(V)$ 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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