Spin-invariant valuations on the octonionic plane

Bernig, Andreas; Voide, Floriane

doi:10.1007/s11856-016-1363-0

Cited by 17 publications

(20 citation statements)

References 41 publications

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“…Various other groups of isometries, also in Riemannian isotropic spaces, have been studied in recent years. Major progress has been made, for instance, in Hermitian integral geometry (in curved spaces), where the interplay between global and local results turned out to be crucial (see [13,14,25,26,79,80,76] and the survey [11]), but various other group actions have been studied successfully as well (see [4,8,9,12,17,18,19,23]).…”

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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“…We use the constructions from Subsection 5.2, and take ω k,n−k λ to correspond to E k Proposition 7.5. For k = n+1 2 and almost every λ, the space of vertical forms in Ω k,n−k −∞ (V × P + (V * )) tr that satisfy g * ω = ψ g (ξ) λ+ n−k 2 ω ∀g ∈ SO + (Q) (28) and a * ω = (−1) n−1 ω is one-dimensional. For large Reλ, this space consists of continuous forms.…”

Section: Classification Of Invariant Continuous Valuationsmentioning

confidence: 99%

Valuation theory of indefinite orthogonal groups

Bernig

Faifman

2017

Journal of Functional Analysis

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Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invariant valuations. As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain-Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations. The operations of pull-back and push-forward by a linear map extend naturally to this class.Comment: 65 pages; Some details in proofs added and minor mistakes corrected, an appendix on wave front sets of G-invariant distributions and a section on the linear algebra of O(p,q) added. Accepted for publication in Journal of Functional Analysi

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“…Alesker's theorem gives the finite-dimensionality of Val G in each of these cases, but the explicit computation of the dimensions required more efforts. This computation was worked out in [1,[9][10][11]17]. The next step is to find a geometrically meaningful basis of Val G .…”

Section: General Backgroundmentioning

confidence: 99%