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Tytuł artykułu

Monotone Valuations on the Space of Convex Functions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider the space Cn of convex functions u defined in Rn with values in R ∪ {∞}, which are lower semi-continuous and such that lim|x| } ∞ u(x) = ∞. We study the valuations defined on Cn which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. We prove integral representations formulas for such valuations which are, in addition, simple or homogeneous.
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2015-03-28
zaakceptowano
2015-06-16
online
2015-07-31
Twórcy
autor
  • Dipartimento di Matematica e Informatica “U.Dini", Viale Morgagni 67/A, 50134, Firenze, Italy
autor
  • Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-857, Japan
Bibliografia
  • [1] W. K. Allard, The Riemann and Lebesgue integrals, lecture notes. Available at: http://www.math.duke.edu~wka/math204/
  • [2] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, New York, 2000.
  • [3] Y. Baryshnikov, R. Ghrist, M. Wright, Hadwiger’s Theorem for definable functions, Adv. Math. 245 (2013), 573-586. [WoS]
  • [4] L. Cavallina, Non-trivial translation invariant valuations on L1, in preparation.
  • [5] A. Colesanti, I. Fragalà, The first variation of the total mass of log-concave functions and related inequalities, Adv. Math. 244 (2013), pp. 708-749.
  • [6] L. C. Evans, R. F. Gariepy, Measure theory and fine properties of fucntions, CRC Press, Boca Raton, 1992.
  • [7] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957.
  • [8] D. Klain, A short proof of Hadwiger’s characterization theorem, Mathematika 42 (1995), 329-339.
  • [9] D. Klain, G. Rota, Introduction to geometric probability, Cambridge University Press, New York, 1997.
  • [10] H. Kone, Valuations on Orlicz spaces and Lφ-star sets, Adv. in Appl. Math. 52 (2014), 82-98.
  • [11] M. Ludwig, Fisher information and matrix-valued valuations, Adv. Math. 226 (2011), 2700-2711. [WoS]
  • [12] M. Ludwig, Valuations on function spaces, Adv. Geom. 11 (2011), 745 - 756. [WoS]
  • [13] M. Ludwig, Valuations on Sobolev spaces, Amer. J. Math. 134 (2012), 824 - 842.
  • [14] M. Ludwig, Covariance matrices and valuations, Adv. in Appl. Math. 51 (2013), 359-366.
  • [15] M. Ober, Lp-Minkowski valuations on Lq-spaces, J. Math. Anal. Appl. 414 (2014), 68-87.
  • [16] T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
  • [17] R. Schneider, Convex bodies: the Brunn-Minkowski theory, second expanded edition, Cambridge University Press, Cambridge, 2014.
  • [18] A. Tsang, Valuations on Lp-spaces, Int. Math. Res. Not. IMRN 2010, 20, 3993-4023.
  • [19] A. Tsang, Minkowski valuations on Lp-spaces, Trans. Amer. Math. Soc. 364 (2012), 12, 6159-6186.
  • [20] T. Wang, Affine Sobolev inequalities, PhD Thesis, Technische Universität, Vienna, 2013.
  • [21] T. Wang, Semi-valuations on BV(Rn), Indiana Univ. Math. J. 63 (2014), 1447–1465.
  • [22] M. Wright, Hadwiger integration on definable functions, PhD Thesis, 2011, University of Pennsylvania.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0012
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