Displaying similar documents to “An example of a 𝒞 1 , 1 function, which is not a d.c. function”

A new proof of Fréchet differentiability of Lipschitz functions

Joram Lindenstrauss, David Preiss (2000)

Journal of the European Mathematical Society

Similarity:

We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on ℓ 2 (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w * closure of the set of its points of Gâteaux differentiability is norm separable.

Biseparating maps on generalized Lipschitz spaces

Denny H. Leung (2010)

Studia Mathematica

Similarity:

Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F-valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are...

Singular points of order k of Clarke regular and arbitrary functions

Luděk Zajíček (2012)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let X be a separable Banach space and f a locally Lipschitz real function on X . For k ∈ ℕ , let Σ k ( f ) be the set of points x ∈ X , at which the Clarke subdifferential ∂ C f ( x ) is at least k -dimensional. It is well-known that if f is convex or semiconvex (semiconcave), then Σ 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,...

A Lipschitz function which is C ∞ on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

Similarity:

We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is C ∞ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...

A note on propagation of singularities of semiconcave functions of two variables

Luděk Zajíček (2010)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in ℝ n propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for n = 2 , these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) ψ ( x ) = ( x , y 1 ( x ) - y 2 ( x ) ) , x ∈ [ 0 , α ] , where y 1 , y 2 are convex and Lipschitz on [ 0 , α ] . In other words: singularities propagate along arcs with finite turn. ...

On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček (1997)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux differentiable bump function. In particular, our result implies the following theorem: If d is a distance function determined by a closed subset A of a Banach space X with a uniformly Gâteaux differentiable norm, then the set of points of X ∖ A at which d is not Gâteaux differentiable is not only a first category set, but...

On continuous composition operators

Wilhelmina Smajdor (2010)

Annales Polonici Mathematici

Similarity:

Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that M ψ : = s u p | | [ r , s , t ; ψ ] | | : r < s < t , r , s , t ∈ I < ∞ , where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove...

Bourgain’s discretization theorem

Ohad Giladi, Assaf Naor, Gideon Schechtman (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

Bourgain’s discretization theorem asserts that there exists a universal constant C ∈ ( 0 , ∞ ) with the following property. Let X , Y be Banach spaces with dim X = n . Fix D ∈ ( 1 , ∞ ) and set δ = e - n C n . Assume that 𝒩 is a δ -net in the unit ball of X and that 𝒩 admits a bi-Lipschitz embedding into Y with distortion at most D . Then the entire space X admits a bi-Lipschitz embedding into Y with distortion at most C D . This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem. ...