RT Journal Article T1 Radial continuous rotation invariant valuations on star bodies A1 Villanueva, Ignacio AB We 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. PB Elsevier SN 0001-8708 YR 2016 FD 2016 LK https://hdl.handle.net/20.500.14352/24365 UL https://hdl.handle.net/20.500.14352/24365 LA eng NO [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. NO Ministerio de Economía y Competitividad (MINECO) NO Comunidad de Madrid DS Docta Complutense RD 5 dic 2023