In: Geometriae Dedicata, 2006, vol. 123, no. 1, p. 153-169
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In: Archiv der Mathematik, 2009, vol. 92, no. 4, p. 314-324
It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension.
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In: Journal of Differential Geometry, 2007, vol. 75, no. 3, p. 433-457
Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on...
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In: Journal für die reine und angewandte Mathematik, 2007, no. 608, p. 1-15
It is shown that the Gromov-Hausdorff limit of a subanalytic 1-parameter family of compact connected sets (endowed with the inner metric) exists. If the family is semialgebraic, then the limit space can be identified with a semialgebraic set over some real closed field. Different notions of tangent cones (pointed Gromov-Hausdorff limits, blow-ups and Alexandrov cones) for a closed connected...
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In: Israel Journal of Mathematics, 2007, vol. 159, no. 1, p. 373-411
An elementary construction of the normal cycle of a compact definable set in Euclidean space (and more generally of a compactly supported constructible function) is given. Here “definable” means definable in some o-minimal structure. The construction is based on the notion of support function and uses only basic o-minimal geometry.
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In: Advances in Mathematics, 2007, vol. 210, no. 2, p. 733-753
Valuations admitting a smooth Crofton formula are studied using Geometric Measure Theory and Rumin's cohomology of contact manifolds. The main technical result is a current representation of a valuation with a smooth Crofton formula. A geometric interpretation of Alesker's product is given for such valuations. As a first application in Finsler geometry, a short proof of the theorem of...
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In: Geometriae Dedicata, 2006, vol. 123, no. 1, p. 153-169
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...
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