Abstract
Properties of local and global slopes of a function and its approximate critical points sets are studied in relation to determination of the function.
References
- L. Ambrosio, N. Gigli, G. Savaré: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel (2008).Google Scholar
- D. Azé: A unified theory for metric regularity of multifunctions, J. Convex Analysis 13/2 (2006) 225–252.https://doi.org/10.68381/jca13023Google Scholar
- D. Azé, J.-N. Corvellec: On some variational properties of metric spaces, J. Fixed Point Theory 5 (2009) 185–200.https://doi.org/10.1007/s11784-008-0054-9Google Scholar
- J. M. Borwein, J. D. Vanderwerff: Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, Cambridge (2010).https://doi.org/10.1017/CBO9781139087322Google Scholar
- T. Z. Boulmezaoud, P. Cieutat, A. Daniilidis: Gradient flows, second-order gradient systems and convexity, SIAM J. Optim. 28/3 (2018) 2049–2066.https://doi.org/10.1137/16M1153418Google Scholar
- F. H. Clarke: Optimization and Nonsmooth Analysis, Wiley, New York (1983).Google Scholar
- R. Correa, P. Gajardo, L. Thibault, D. Zagrodny: Existence of minimizers on drops, SIAM J. Optim. 23 (2013) 1154–1166.https://doi.org/10.1137/120875934Google Scholar
- A. Daniilidis: Personal communication, Borovets, July 2023.Google Scholar
- A. Daniilidis, T. M. Le, D. Salas: Metric compatibility and determination in complete metric spaces, Math. Zeitschrift 308/4 (2024), art.no. 62, 31p.https://doi.org/10.1007/s00209-024-03609-2Google Scholar
- A. Daniilidis, D. Salas: A determination theorem in terms of the metric slope, Proc. Amer. Math. Soc. 150 (2022) 4325–4333.Google Scholar
- E. De Giorgi, A. Marino, M. Tosques: Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 8/3 (1980) 180–187.Google Scholar
- D. Drusvyatskiy, A. D. Ioffe, A. S. Lewis: Curves of descent, SIAM J. Control Optim. 53/1 (2015) 114–138.https://doi.org/10.1137/130920216Google Scholar
- Y. Giga, N. Hamamuki, A. Nakayasu: Eikonal equations in metric spaces, Trans. Amer. Math. Soc. 367/1 (2015) 49–66.https://doi.org/10.1090/S0002-9947-2014-05893-5Google Scholar
- A. Göpfert, H. Riahi, C. Tammer, C. Zălinescu: Variational Methods in Partially Ordered Spaces, Springer, New York (2003).Google Scholar
- A. Ioffe: Variational Analysis of Regular Mappings. Theory and Applications, Springer Monographs in Mathematics, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
- N. Ivanov: Personal communication, Chernomoretz (2023).Google Scholar
- M. Ivanov, N. Zlateva: A new proof of the integrability of the subdifferential of a convex function on a Banach space, Proc. Amer. Math. Soc. 136/5 (2008) 1787–1793.https://doi.org/10.1090/S0002-9939-08-09178-8Google Scholar
- A. A. Khan, C. Tammer, C. Zalinescu: Set-Valued Optimization: An Introduction with Applications, Springer, Berlin (2015).https://doi.org/10.1007/978-3-642-54265-7Google Scholar
- M. Konstantinov, N. Zlateva: Epsilon subdifferential method and integrability, J. Convex Analysis 29/2 (2022) 571-582.https://doi.org/10.68381/jca29033Google Scholar
- Q. Liu, N. Shanmugalingam, X. Zhou: Equivalence of solutions of eikonal equation in metric spaces, J. Diff. Equations 272 (2021) 979–1014.https://doi.org/10.1016/j.jde.2020.10.018Google Scholar
- J. J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
- M. Muratori, G. Savaré: Gradient flows and evolution variational inequalities in metric spaces. I: Structural properties, J. Funct. Anal. 278/4 (2020), art. no. 108347, 67p.https://doi.org/10.1016/j.jfa.2019.108347Google Scholar
- P. Pérez-Aros, D. Salas, E. Vilches: Determination of convex functions via subgradients of minimal norm, Math. Programming A 190/1-2 (2021) 561–583.https://doi.org/10.1007/s10107-020-01550-wGoogle Scholar
- R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd ed., Lecture Notes in Mathematics 1364, Springer, Berlin (1993).Google Scholar
- R. T. Rockafellar: Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966) 497–510.https://doi.org/10.2140/pjm.1966.17.497Google Scholar
- R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
- R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
- R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften Vol. 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
- L. Thibault, D. Zagrodny: Determining functions by slopes, Comm. Contemp. Math. 25/7 (2023), art. no. 225014, 30p.https://doi.org/10.1142/S0219199722500146Google Scholar
Suggested citation
Published by Heldermann Verlag, 2025. Rights now held by Banach Press.
