Welcome To WUJNS
武汉大学学报 英文版 | Wuhan University Journal of Natural Sciences
Wan Fang
CNKI
CSCD
Wuhan University
Latest Article
Valuations on Concave Functions and Log-Concave Functions
Time:2019-11-15  
LIU Lijuan
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China
Abstract:
Recently, the theory of valuations on function spaces has been rapidly growing. It is more general than the classical theory of valuations on convex bodies. In this paper, all continuous, SL(n) and translation invariant valuations on concave functions and log-concave functions are completely classified, respectively.
Key words:valuations; concave functions; log-concave functions; characterization theorem
CLC number:O 184
References:
[1]	Ludwig M. Fisher information and matrix-valued valuations [J]. Adv Math, 2011, 226 (3): 2700-2711.
[2]	Ludwig M. Valuations on Sobolev spaces [J]. Amer J Mat, 2012, 134 (3): 827-842.
[3]	Ma D. Real-valued valuations on Sobolev spaces [J]. Sci China Mat, 2016, 59 (5): 921-934.
[4]	Ludwig M. Covariance matrices and valuations [J]. Adv in Appl Math, 2013, 51: 359-366.
[5]	Ober M.  -Minkowski valuations on  -spaces [J]. J Math Anal Appl, 2014, 414 (1): 68-87.
[6]	Tsang A. Valuations on  -spaces [J]. Int Math Res Not, 2010, 20:3993-4023.
[7]	Tsang A. Minkowski valuations on  -spaces [J]. Trans Amer Math Soc, 2012, 364 (12): 6159-6186.
[8]	Baryshnikov Y, Ghrist R, Wright M. Hadwiger’s theorem for definable functions [J]. Adv Math, 2013, 245: 573-586.
[9]	Alesker S. Valuations on convex functions and convex sets and Monge-Ampère operators [EB/OL]. [2017-04-02]. http: //arxiv.org/pdf/1703.08778.pdf.
[10]	Cavallina L, Colesanti A.  Monotone valuations on the space of convex functions [J]. Anal Geom Metr Spaces, 2015, 3 (1): 167-211.
[11]	Colesanti A, Ludwig M, Mussnig F. Valuations on convex functions [J]. Int Math Res Not IMRN, Oxford: Oxford Aca-demic, 2018, DOI: https://doi.org/10.1093/imrn/rnx189.
[12]	Colesanti A, Ludwig M, Mussnig F. Minkowski valuations on convex functions [J]. Cala Var Partial Differential Equa-tions, 2017, 56:162.
[13]	Mussnig F. Valuations on log-concave functions [EB/OL]. [2017-07-20]. https://arXiv.org/pdf/1707.06428.
[14]	Colesanti A, Lombardi N. Valuations on the space of qua-si-concave functions [C] // Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics 2169. Cham: Springer International Publishing, 2017: 71-105.
[15]	Colesanti A, Lombardi N, Parapatits L. Translation invariant valuations on quasi-concave functions [J]. Studia Mathemat-ica, 2018, 243:79-99.
[16]	Wang T. Semi-valuations on  [J]. Indiana Univ Math J, 2014, 63 (5): 1447-1465.
[17]	Dal Maso G. An Introduction to  -Convergence, Progress in Nonlinear Differential Equations and Their Applications [M]. Boston: Birkhauser Inc, 1993.
Welcome To WUJNS

HOME | Aim and Scope | Editoral Board | Current Issue | Back Issue | Subscribe | Crosscheck | Polishing | Contact us Copyright © 1997-2019 All right reserved