Czasopismo
Tytuł artykułu
Autorzy
Warianty tytułu
Języki publikacji
Abstrakty
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.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
Numer
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