# Article

 Title: Singular points of order $k$ of Clarke regular and arbitrary functions (English) Author: Zajíček, Luděk Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 53 Issue: 1 Year: 2012 Pages: 51-63 Summary lang: English . Category: math . Summary: Let $X$ be a separable Banach space and $f$ a locally Lipschitz real function on $X$. For $k\in \mathbb N$, let $\Sigma_k(f)$ be the set of points $x\in X$, at which the Clarke subdifferential $\partial^Cf(x)$ is at least $k$-dimensional. It is well-known that if $f$ is convex or semiconvex (semiconcave), then $\Sigma_k(f)$ can be covered by countably many Lipschitz surfaces of codimension $k$. We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove also two results on arbitrary functions, which work with Hadamard directional derivatives and can be considered as generalizations of our theorem on $\Sigma_k(f)$ of Clarke regular functions (since each of them easily implies this theorem). (English) Keyword: Clarke regular functions Keyword: singularities Keyword: Hadamard derivative MSC: 26B25 MSC: 49J52 idZBL: Zbl 1249.49021 idMR: MR2880910 . Date available: 2012-02-07T10:23:17Z Last updated: 2014-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/141825 . Reference: [1] Albano P., Cannarsa P.: Singularities of semiconcave functions in Banach spaces.in: Stochastic Analysis, Control, Optimization and Applications, W.M. McEneaney, G.G. Yin, Q. Zhang (eds.), Birkhäuser, Boston, 1999, pp. 171–190. Zbl 0923.49010, MR 1702959 Reference: [2] Alberti G., Ambrosio L., Cannarsa P.: On the singularities of convex functions.Manuscripta Math. 76 (1992), 421–435. Zbl 0784.49011, MR 1185029, 10.1007/BF02567770 Reference: [3] Anderson R.D., Klee V.L., Jr.: Convex functions and upper semi-continuous collections.Duke Math. J. 19 (1952), 349–357. Zbl 0047.15702, MR 0047346, 10.1215/S0012-7094-52-01935-2 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, MA, 2004. Zbl 1095.49003, MR 2041617 Reference: [5] Duda J., Zajíček L.: Smallness of singular sets of semiconvex functions in separable Banach spaces.submitted. Reference: [6] Ioffe A.D.: Typical convexity (concavity) of Dini-Hadamard upper (lower) directional derivatives of functions on separable Banach spaces.J. Convex Anal. 17 (2010), 1019–1032. Zbl 1208.46043, MR 2731290 Reference: [7] Kato T.: Perturbation Theory for Linear Operators.Springer, Berlin, 1976. Zbl 0836.47009, MR 0407617 Reference: [8] Nagai H.V., Luc D.T., Théra M.: Approximate convex functions.J. Nonlinear Convex Anal. 1 (2000), 155–176. MR 1777137 Reference: [9] Nekvinda A., Zajíček L.: Gâteaux differentiability of Lipschitz functions via directional derivatives.Real Anal. Exchange 28 (2002-2003), 287–320. MR 2009756 Reference: [10] Pavlica D.: On the points of non-differentiability of convex functions.Comment. Math. Univ. Carolin. 45 (2004), 727–734. Zbl 1100.26006, MR 2103086 Reference: [11] Preiss D.: Almost differentiability of convex functions in Banach spaces and determination of measures by their values on balls.Collection: Geometry of Banach spaces (Strobl, 1989), London Math. Soc. Lecture Note Ser. 158, Cambridge University Press, Cambridge, 1990, pp. 237–244. Zbl 0758.46034, MR 1110199 Reference: [12] Zajíček L.: On the differentiation of convex functions in finite and infinite dimensional spaces.Czechoslovak Math. J. 29 (1979), 340–348. MR 0536060 Reference: [13] Zajíček L.: Differentiability of the distance function and points of multi-valuedness of the metric projection in Banach space.Czechoslovak Math. J. 33 (108) (1983), 292–308. MR 0699027 Reference: [14] Zajíček L.: On Lipschitz and D.C. surfaces of finite codimension in a Banach space.Czechoslovak Math. J. 58 (133) (2008), 849–864. MR 2455942, 10.1007/s10587-008-0055-2 .

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