| Title:
|
Differences of two semiconvex functions on the real line (English) |
| Author:
|
Kryštof, Václav |
| Author:
|
Zajíček, Luděk |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
21-37 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is proved that real functions on $\mathbb R$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $\mathbb R$ (i.e. those locally Lipschitz functions on $\mathbb R$ for which $f'_+(x) = \lim_{t \to x+} f'_+(t)$ and $f'_-(x) = \lim_{t \to x-} f'_-(t)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DSC_{\omega}$ of functions on $\mathbb R$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new notion of $[\omega]$-variation. We prove that $f \in DSC_{\omega}$ if and only if $f$ is continuous and there exists $D>0$ such that $f'_+$ has locally finite $[D \omega]$-variation. This result is proved via a generalization of the classical Jordan decomposition theorem which characterizes the differences of two $\omega$-nondecreasing functions (defined by the inequality $f(y) \geq f(x)- \omega(y-x)$ for $y>x$) on $[a,b]$ as functions with finite $[2\omega]$-variation. The research was motivated by a recent article by J. Duda and L. Zajíček on Gâteaux differentiability of semiconvex functions, in which surfaces described by differences of two semiconvex functions naturally appear. (English) |
| Keyword:
|
semiconvex function with general modulus |
| Keyword:
|
difference of two semiconvex functions |
| Keyword:
|
$\omega$-nondecreasing function |
| Keyword:
|
$[\omega]$-variation |
| Keyword:
|
regulated function |
| MSC:
|
26A45 |
| MSC:
|
26A48 |
| MSC:
|
26A51 |
| MSC:
|
26B05 |
| idZBL:
|
Zbl 06562193 |
| idMR:
|
MR3478336 |
| DOI:
|
10.14712/1213-7243.2015.153 |
| . |
| Date available:
|
2016-04-12T05:01:37Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144912 |
| . |
| Reference:
|
[1] Benyamini Y., Lindenstrauss J.: Geometric Nonlinear Functional Analysis.Vol. 1, Colloquium Publications, 48, American Mathematical Society, Providence, 2000. Zbl 0946.46002, MR 1727673 |
| Reference:
|
[2] Bruckner A.: Differentiation of Real Functions.CRM Monograph Series, 5, American Mathematical Society, Providence, RI, 1994. MR 1274044 |
| Reference:
|
[3] Correa R., Jofré A.: Tangentially continuous directional derivatives in nonsmooth analysis.J. Optim. Theory Appl. 61 (1989), 1–21. MR 0993912, 10.1007/BF00940840 |
| Reference:
|
[4] Cannarsa P., Sinestrari C.: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control.Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston, 2004. Zbl 1095.49003, MR 2041617 |
| Reference:
|
[5] Duda J., Zajíček L.: Semiconvex functions: representations as suprema of smooth functions and extensions.J. Convex Anal. 16 (2009), 239–260. MR 2531202 |
| Reference:
|
[6] Duda J., Zajíček L.: Smallness of singular sets of semiconvex functions in separable Banach spaces.J. Convex Anal. 20 (2013), 573–598. MR 3098482 |
| Reference:
|
[7] Fabian M.: Gâteaux Differentiability of Convex Functions and Topology. Weak Asplund Spaces.Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1997. Zbl 0883.46011, MR 1461271 |
| Reference:
|
[8] Fraňková D.: Regulated functions.Math. Bohem. 116 (1991), 20–59. MR 1100424 |
| Reference:
|
[9] Goffman C., Moran G., Waterman D.: The structure of regulated functions.Proc. Amer. Math. Soc. 57 (1976), 61–65. MR 0401993, 10.1090/S0002-9939-1976-0401993-5 |
| Reference:
|
[10] Jourani A., Thibault L., Zagrodny D.: $C^{1,\omega(\cdot)}$-regularity and Lipschitz-like properties of subdifferential.Proc. London Math. Soc. 105 (2012), 189–223. MR 2948792 |
| Reference:
|
[11] Mifflin R.: Semismooth and semiconvex functions in constrained optimization.SIAM J. Control Optimization 15 (1977), 959–972. Zbl 0376.90081, MR 0461556, 10.1137/0315061 |
| Reference:
|
[12] Nagai H.V., Luc D.T., Théra M.: Approximate convex functions.J. Nonlinear Convex Anal. 1 (2000), 155–176. MR 1777137 |
| Reference:
|
[13] Spingarn J. E.: Submonotone subdifferentials of Lipschitz functions.Trans. Amer. Math. Soc. 264 (1981) 77–89. Zbl 0465.26008, MR 0597868, 10.1090/S0002-9947-1981-0597868-8 |
| . |
