Abstract
A deformation is a positive continuous function defined on an appropriate interval. Through deformations, we generalize the notion of concavity for functions. We introduce the order function of a deformation, which permits to determine precisely the ranking of a deformation by taking account of the corresponding concavities. In the hierarchy, the action of positive constant multiples provides an equivalence relation and, if we focus on \(C^1\)-deformations, a one-to-one correspondence between the equivalence classes and the order functions is determined. Deformations having a constant valued order function play a fundamental role, and this is only the case of power functions. We show that the concavity associated to a deformation whose order function is nonincreasing and uniformly bounded from above by \(q\in {\mathbb {R}}\) can approximate the concavity associated to the power function of exponent q. Finally, we review three examples of deformations whose order function is nonincreasing and uniformly bounded from above. One is a power function, and the others are related to a concavity preserved by the Dirichlet heat flow in convex domains of Euclidean space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
References
Avriel, M., Zang, I.: Generalized convex functions with applications to nonlinear programming. In: Mathematical Programs for Activity Analysis, pp. 23–33. Rep. Belgo-Israeli Colloq. Operations Res., Pope Hadrian VI College, Univ. Louvain, Louvain (1974)
Bogachev, V.I.: Measure Theory, vol. I, p. 500. Springer, Heidelberg (2007)
Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)
Ishige, K., Salani, P., Takatsu, A.: To logconcavity and beyond. Commun. Contemp. Math. 22(2), 1950009–17 (2020)
Ishige, K., Salani, P., Takatsu, A.: New characterizations of log-concavity via Dirichlet heat flow. Ann. Mat. Pura Appl. 201(4), 1531–1552 (2022)
Ishige, K., Salani, P., Takatsu, A.: Characterization of \(F\)-concavity preserved by the Dirichlet heat flow. arXiv:2207.13449
Kannan, R., Krueger, C.K.: Advanced Analysis on the Real Line, p. 259. Springer, NY (1996)
Matsuzoe, H., Takatsu, A.: Gauge freedom of entropies on \(q\)-Gaussian measures. In: Progress in Information Geometry–Theory and Applications, pp. 127–152. Springer, Cham (2021)
Naudts, J.: Generalised Thermostatistics. Springer, London (2011)
Ohta, S.-I., Takatsu, A.: Displacement convexity of generalized relative entropies. II. Comm. Anal. Geom. 21(4), 687–785 (2013)
Pal, S., Wong, T.-K.L.: Exponentially concave functions and a new information geometry. Ann. Probab. 46(2), 1070–1113 (2018)
Shioya, T.: Metric Measure Geometry. IRMA Lectures in Mathematics and Theoretical Physics, vol. 25, p. 182. EMS Publishing House, Zürich (2016)
Wong, T.-K.L., Yang, J.: Logarithmic divergences geometry and interpretation of curvature. In: Geometric Science of Information. Lecture Notes in Computer Science, vol. 11712, pp. 413–422. Springer, Cham (2019)
Wong, T.-K.L., Zhang, J.: Tsallis and Rényi deformations linked via a new \(\lambda \)-duality. IEEE Trans. Inform. Theory 68(8), 5353–5373 (2022)
Acknowledgements
K. I. and A. T. were supported in part by JSPS KAKENHI Grant Number 19H05599. P. S. was supported in part by INdAM through a GNAMPA Project. A. T. was supported in part by JSPS KAKENHI Grant Number 19K03494.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Communicated by Shinto Eguchi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ishige, K., Salani, P. & Takatsu, A. Hierarchy of deformations in concavity. Info. Geo. 7 (Suppl 1), 251–269 (2024). https://doi.org/10.1007/s41884-022-00088-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41884-022-00088-4