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Radial continuous rotation invariant valuations on star bodies

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dc.contributor.authorVillanueva, Ignacio
dc.date.accessioned2023-06-18T06:50:24Z
dc.date.available2023-06-18T06:50:24Z
dc.date.issued2016
dc.description.abstractWe characterize the positive radial continuous and rotation invariant valuations V defined on the star bodies of Rn as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is,V(K)=∫Sn−1θ(ρK)dm, where θ is a positive continuous function, ρK is the radial function associated to K and m is the Lebesgue measure on Sn−1. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.refereedTRUE
dc.description.sponsorshipMinisterio de Economía y Competitividad (MINECO)
dc.description.sponsorshipComunidad de Madrid
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/36032
dc.identifier.citation[1] S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. 149 (1999), 977–1005. [2] S. Alesker, Erratum: Rotation invariant valuations on convex sets, Ann. of Math., 166 (2007), 947–948. [3] S. Alesker, Theory of valuations on manifolds: a urvey, Geom. Funct. Anal. 17 (2007), 1321–1341. [4] D. Cohn, Measure Theory, Birkhauser 1980. [5] R. J. Gardner, A positive answer to the BusemannPetty problem in three dimensions, Ann. of Math. (2) 140 (1994), 435-447. [6] R. J. Gardner, Geometric Tomography, second edition, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 2006. [7] R. J. Gardner, A. Koldobsky, T. Schlumprecht, An analytical solution to the BusemannPetty problem on sections of convex bodies, Ann. of Math. (2) 149 (1999) 691-703. [8] H. Hadwiger, Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, SpringerVerlag, New York, 1957. [9] P. Halmos, Measure Theory, Springer-Verlag, New York, 1974. [10] C. H. Jimenez, I. Villanueva, Characterization of dual mixed volumes via polymeasures, J. Math. Anal. Appl. 426 (2015) 688–699. [11] A. G. Khovanskii, A. V. Pukhlikov, Finitely additive measures on virtual polyhedra (Russian), Algebra i analiz. 4 (1992), 161-85; translation in St. Petersburg Math. J.4 (1993), 337-356. [12] D. A. Klain, Star Valuations and Dual Mixed Volumes, Adv. Math. 121 (1996), 80–101. [13] D. A. Klain, Invariant Valuations on Star-Shaped Sets, Adv. Math. 125 (1997), 95–113. [14] H. Kone, Valuations on Orlicz spaces and L ϕ-star sets, Adv. in Appl. Math. 52 (2014), 8298 [15] M. Ludwig, Intersection bodies and valuations, American Journal of Mathematics, 128 (2006), 1409–1428. [16] M. Ludwig, M. Reitzner, A classification of SL(n) invariant valuations, Ann. of Math. 172 (2010), 1219-267. [17] E. Lutwak, Dual Mixed volumes, Pacific J. Math 58 (1975), 531-538. [18] D. Ma, Real-valued valuations on Sobolev spaces, arXiv:1505.02004 [19] A. Tsang, Valuations on L p -Spaces (Int. Math. Res. Not. 20 (2010), 39934023) [20] G. Zhang, A positive answer to the BusemannPetty problem in four dimensions, Ann. of Math. (2) 149 (1999) 535-543.
dc.identifier.doi10.1016/j.aim.2015.12.030
dc.identifier.issn0001-8708
dc.identifier.officialurlhttp://www.sciencedirect.com/science/article/pii/S0001870816000335
dc.identifier.relatedurlhttp://arxiv.org/abs/1503.06064
dc.identifier.relatedurlhttp://www.sciencedirect.com
dc.identifier.urihttps://hdl.handle.net/20.500.14352/24365
dc.issue.number19
dc.journal.titleAdvances in Mathematics
dc.language.isoeng
dc.page.final981
dc.page.initial961
dc.publisherElsevier
dc.relation.projectIDMTM2014-54240-P
dc.relation.projectIDQUITEMAD+ (S2013/ICE-2801)
dc.rights.accessRightsrestricted access
dc.subject.cdu517
dc.subject.keywordConvex geometry
dc.subject.keywordValuations
dc.subject.keywordStar bodies
dc.subject.ucmAnálisis matemático
dc.subject.unesco1202 Análisis y Análisis Funcional
dc.titleRadial continuous rotation invariant valuations on star bodies
dc.typejournal article
dc.volume.number291
dspace.entity.typePublication
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