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Valuations on Manifolds and Integral Geometry

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Abstract

We construct new operations of pull-back and push-forward on valuations on manifolds with respect to submersions and immersions. A general Radon-type transform on valuations is introduced using these operations and the product on valuations. It is shown that the classical Radon transform on smooth functions, and the well-known Radon transform on constructible functions, with respect to the Euler characteristic, are special cases of this new Radon transform. An inversion formula for the Radon transform on valuations has been proven in a specific case of real projective spaces. Relations of these operations to yet another classical type of integral geometry, Crofton and kinematic formulas, are indicated.

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Authors and Affiliations

  1. Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Semyon Alesker

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  1. Semyon Alesker
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Correspondence to Semyon Alesker.

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Partially supported by ISF grant 701/08.

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Alesker, S. Valuations on Manifolds and Integral Geometry. Geom. Funct. Anal. 20, 1073–1143 (2010). https://doi.org/10.1007/s00039-010-0088-1

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