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Continuous valuations on the space of Lipschitz on the sphere

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dc.contributor.authorColesanti, Andrea
dc.contributor.authorPagnini, Daniele
dc.contributor.authorTradacete Pérez, Pedro
dc.contributor.authorVillanueva, Ignacio
dc.date.accessioned2023-06-17T08:29:00Z
dc.date.available2023-06-17T08:29:00Z
dc.date.issued2020-11-30
dc.description.abstractWe study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere Sn−1. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere. Contents
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.refereedFALSE
dc.description.sponsorshipMinisterio de Ciencia e Innovación (MICINN)/FEDER
dc.description.sponsorshipMinisterio de Economía y Competitividad (MINECO)
dc.description.sponsorshipComunidad de Madrid
dc.description.sponsorshipCentro de Excelencia Severo Ochoa
dc.description.sponsorshipUniversidad Complutense de Madrid
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/74378
dc.identifier.citation[1] S. Alesker, Continuous Rotation Invariant Valuations on Convex Sets. Ann. of Math.149 (1999), 977-1005. [2] R. B. Ash, Probability and Measure Theory (Second Edition). Academic Press, San Diego, 1999. [3] K. P. S. Bhaskara Rao, M. Bhaskara Rao, Theory of Charges. A Study of Finitely Additive Measures. With a foreword by D. M. Stone. Pure and Applied Mathematics, 109. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. [4] L. Cavallina, A. Colesanti, Monotone Valuations on the Space of Convex Functions. Anal. Geom. Metr. Spaces 3 (2015), 167-211. [5] D. L. Cohn, Measure Theory. Reprint of the 1980 original. Birkh¨auser Boston, Inc., Boston, MA, 1993. [6] A. Colesanti, N. Lombardi, L. Parapatits, Translation Invariant Valuations on the Space of Quasi-Concave Functions. Studia Math. 243 (2018), no. 1, 79-99. [7] A. Colesanti, M. Ludwig, F. Mussnig, Valuations on Convex Functions. Int. Math. Res. Not. IMRN 2019, no. 8, 2384-2410. [8] A. Colesanti, D. Pagnini, P. Tradacete, I. Villanueva, A Class of Invariant Valuations on Lip(Sn−1). Adv. Math. 366, 37 pages, 2020. [9] N. Dunford, J. T. Schwartz. Linear Operators, part I, General Theory. Interscience Publishers, 1958. [10] L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, 1992. [11] H. Hadwiger, Vorlesungen über Inhalt, Oberfl¨ache und Isoperimetrie. Springer, Berlin, 1957. [12] D. A. Klain, Star Valuations and Dual Mixed Volumes. Adv. Math. 121 (1996), no. 1, 80-101. [13] D. A. Klain, Invariant Valuations on Star-Shaped Sets. Adv. Math. 125 (1997), no. 1, 95-113. [14] D. A. Klain, G.-C. Rota, Introduction to Geometric Probability. Cambridge University Press, Cambridge, 1997. [15] M. Ludwig, Valuations on Function Spaces, Adv. Geom. 11 (2011), 745-756. [16] M. Ludwig, Fisher Information and Matrix-Valued Valuations. Adv. Math. 226 (2011), no. 3, 27002711. [17] M. Ludwig, Valuations on Sobolev Spaces. Amer. J. Math. 134 (2012), 827-842. [18] D. Ma, Analysis of Sobolev Spaces in the Context of Convex Geometry and Investigations of the Busemann-Petty Problem. PhD Thesis, TU Wien, 2015. [19] E. J. McShane, Extension of Range of Functions. Bull. Amer. Math. Soc. 40 (1934), 837-842. [20] F. Mussnig, Volume, Polar Volume and Euler Characteristic for Convex Functions. Adv. Math. 344 (2019), 340-373. [21] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory (Second Expanded Edition). Cambridge University Press, London, 2014. [22] P. Tradacete, I. Villanueva, Radial Continuous Valuations on Star Bodies. J. Math.Anal. Appl. 454 (2017), 995-1018. [23] P. Tradacete, I. Villanueva, Continuity and Representation of Valuations on StarBodies. Adv. Math. 329 (2018), 361-391. [24] P. Tradacete, I. Villanueva, Valuations on Banach Lattices. Int. Math. Res. Not.IMRN 2020, no. 1, 287-319. [25] A. Tsang, Valuations on Lp-Spaces. Int. Math. Res. Not. IMRN, 20 (2010), 3993-4023. [26] A. Tsang, Minkowski Valuations on Lp-Spaces. Trans. Amer. Math. Soc. 364 (2012), no. 12, 6159-6186. [27] I. Villanueva, Radial Continuous Rotation Invariant Valuations on Star Bodies. Adv. Math. 291 (2016), 961-981. [28] N. Weaver, Lipschitz Algebras. Second Edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018.
dc.identifier.doi10.1016/j.jfa.2020.108873
dc.identifier.issn0022-1236
dc.identifier.officialurlhttps://doi.org/10.1016/j.jfa.2020.108873
dc.identifier.urihttps://hdl.handle.net/20.500.14352/7253
dc.issue.number4
dc.journal.titleJournal of functional analysis
dc.language.isoeng
dc.publisherElsevier
dc.relation.projectIDMTM2016-76808-P
dc.relation.projectIDMTM2017-88385-P
dc.relation.projectIDQUITEMAD+-CM (S2013/ICE-2801)
dc.relation.projectID(SEV-2015-0554; 20205CEX001)
dc.relation.projectIDUCM (910346)
dc.rights.accessRightsopen access
dc.subject.cdu517
dc.subject.keywordGeometric valuation Theory
dc.subject.keywordLipschitz functions
dc.subject.keywordIntegral representation
dc.subject.ucmAnálisis matemático
dc.subject.unesco1202 Análisis y Análisis Funcional
dc.titleContinuous valuations on the space of Lipschitz on the sphere
dc.typejournal article
dc.volume.number280
dspace.entity.typePublication
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