American Institute of Mathematical Sciences

Advanced
January  2008, 7(1): 119-124. doi: 10.3934/cpaa.2008.7.119

Refinement of the Benoist theorem on the size of Dini subdifferentials

Ludovic Rifford 1,

1. 

Université de Nice-Sophia Antipolis, Laboratoire J.A. Dieudonné, Parc Valrose, 06108 Nice Cedex 02, France

Received  September 2006 Revised  May 2007 Published  October 2007

Given a lower semicontinuous function $f:\mathbb R^n \rightarrow \mathbb R \cup$ {$+\infty$}, we prove that the set of points of $\mathbb R^n$ where the lower Dini subdifferential has convex dimension $k$ is countably $(n-k)$-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.
Keywords: Nonsmooth analysis, generalized differentials..
Mathematics Subject Classification: Primary: 49J52, 26B35; Secondary: 26BX.
Citation: Ludovic Rifford. Refinement of the Benoist theorem on the size of Dini subdifferentials. Communications on Pure & Applied Analysis, 2008, 7 (1) : 119-124. doi: 10.3934/cpaa.2008.7.119
[1]

Mohamed Aly Tawhid. Nonsmooth generalized complementarity as unconstrained optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 411-423. doi: 10.3934/jimo.2010.6.411
[2]

Vladimir F. Demyanov, Julia A. Ryabova. Exhausters, coexhausters and converters in nonsmooth analysis. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1273-1292. doi: 10.3934/dcds.2011.31.1273
[3]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[4]

Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 559-571. doi: 10.3934/dcds.2011.30.559
[5]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389
[6]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5337-5354. doi: 10.3934/dcdsb.2020346
[7]

Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135
[8]

Ferrán Valdez. Veech groups, irrational billiards and stable abelian differentials. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1055-1063. doi: 10.3934/dcds.2012.32.1055
[9]

Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254
[10]

Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1
[11]

Jia Cai, Junyi Huo. Sparse generalized canonical correlation analysis via linearized Bregman method. Communications on Pure & Applied Analysis, 2020, 19 (8) : 3933-3945. doi: 10.3934/cpaa.2020173
[12]

Marzia Bisi, Maria Paola Cassinari, Maria Groppi. Qualitative analysis of the generalized Burnett equations and applications to half--space problems. Kinetic & Related Models, 2008, 1 (2) : 295-312. doi: 10.3934/krm.2008.1.295
[13]

Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial & Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611
[14]

Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2022, 11 (1) : 259-282. doi: 10.3934/eect.2021002
[15]

Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3433-3457. doi: 10.3934/dcds.2012.32.3433
[16]

Jonathan Chaika, Yitwah Cheung, Howard Masur. Winning games for bounded geodesics in moduli spaces of quadratic differentials. Journal of Modern Dynamics, 2013, 7 (3) : 395-427. doi: 10.3934/jmd.2013.7.395
[17]

Anton Zorich. Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials. Journal of Modern Dynamics, 2008, 2 (1) : 139-185. doi: 10.3934/jmd.2008.2.139
[18]

Julien Grivaux, Pascal Hubert. Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum. Journal of Modern Dynamics, 2014, 8 (1) : 61-73. doi: 10.3934/jmd.2014.8.61
[19]

Thomas Kappeler, Yannick Widmer. On nomalized differentials on spectral curves associated with the sinh-Gordon equation. Journal of Geometric Mechanics, 2021, 13 (1) : 73-143. doi: 10.3934/jgm.2020023
[20]

Anna M. Barry, Esther WIdiasih, Richard Mcgehee. Nonsmooth frameworks for an extended Budyko model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2447-2463. doi: 10.3934/dcdsb.2017125

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy