Algebraic Integral Geometry

Bernig, Andreas

doi:10.1007/978-3-642-22842-1_5

Cited by 34 publications

(38 citation statements)

References 50 publications

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“…9.1]). The bilinear structure of the right side of the kinematic formula (1) motivated the introduction and study of kinematic operators, in the recently developed field of algebraic integral geometry, which led to new insights, generalizations, and to a profound understanding of the structure of integral geometric formulae (see [10], [27]) in connection with the algebraic structure of translation invariant valuations (see [3], also for further references).…”

Section: Introductionmentioning

confidence: 99%

Kinematic formulae for tensorial curvature measures

Hug

Weis

2018

Annali di Matematica

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Tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. We prove a complete set of kinematic formulae for such tensorial curvature measures on convex bodies and for their (nonsmooth) generalizations on convex polytopes. These formulae express the integral mean of the tensorial curvature measure of the intersection of two given convex bodies (resp. polytopes), one of which is uniformly moved by a proper rigid motion, in terms of linear combinations of tensorial curvature measures of the given convex bodies (resp. polytopes). We prove these results in a more direct way than in the classical proof of the principal kinematic formula for curvature measures, which uses the connection to Crofton formulae to determine the involved constants explicitly.Date: December 26, 2016. 2010 Mathematics Subject Classification. Primary: 52A20, 53C65; secondary: 52A22, 52A38, 28A75.

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Section: Introductionmentioning

confidence: 99%

Kinematic formulae for tensorial curvature measures

Hug

Weis

2018

Annali di Matematica

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“…Note that Area 0 is simply the space of smooth signed measures and M r , the r-th moment map, is not injective on this space, which is why we have to assume that k ≥ 1 in Part (2).…”

Section: Theoremmentioning

confidence: 99%

Dual area measures and local additive kinematic formulas

Bernig

2019

Geom Dedicata

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We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map.Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natural convolution product which encodes the local additive kinematic formulas for groups acting transitively on the unit sphere.As an application of this new integral-geometric structure, we obtain the local additive kinematic formulas in hermitian vector spaces in a very explicit way.2000 Mathematics Subject Classification. 53C65, 52A22.

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“…Similarly to the real-valued case, as observed by Alesker and Bernig (private communication), any Z ∈ (Val⊗V) G , with G a closed subgroup G ⊂ O(n) acting transitively on S n−1 , is smooth. For the convenience of the reader, we give a proof following the arguments of Corollary 3.3 in Fu [24] and Theorem 4.1 in Bernig [16].…”

Section: Definition 33 We Say That a Valuationmentioning

confidence: 99%

SL(m,C)-equivariant and translation covariant continuous tensor valuations

Abardia-Evéquoz¹,

Böröczky²,

Domokos³

et al. 2019

Journal of Functional Analysis

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The space of continuous, SL(m, C)-equivariant, m ≥ 2, and translation covariant valuations taking values in the space of real symmetric tensors on C m ∼ = R 2m of rank r ≥ 0 is completely described. The classification involves the moment tensor valuation for r ≥ 1 and is analogous to the known classification of the corresponding tensor valuations that are SL(2m, R)equivariant, although the method of proof cannot be adapted. * AMS 2010 subject classification: Primary 52B45; Secondary 52A40

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Algebraic Integral Geometry

Abstract: Abstract. A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.

Cited by 34 publications

References 50 publications

Kinematic formulae for tensorial curvature measures

Kinematic formulae for tensorial curvature measures

Dual area measures and local additive kinematic formulas

SL(m,C)-equivariant and translation covariant continuous tensor valuations

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