Displaying similar documents to “On Fréchet differentiability of convex functions on Banach spaces”

On sets of non-differentiability of Lipschitz and convex functions

Luděk Zajíček (2007)

Mathematica Bohemica

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We observe that each set from the system 𝒜 ˜ (or even 𝒞 ˜ ) is Γ -null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ℝ n is σ -strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous...

Topological classification of closed convex sets in Fréchet spaces

Taras Banakh, Robert Cauty (2011)

Studia Mathematica

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We prove that each non-separable completely metrizable convex subset of a Fréchet space is homeomorphic to a Hilbert space. This resolves a more than 30 years old problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowolski and Toruńczyk, this result implies that each closed convex subset of a Fréchet space is homeomorphic to [ 0 , 1 ] ⁿ × [ 0 , 1 ) m × ℓ ₂ ( κ ) for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.

The distance between subdifferentials in the terms of functions

Libor Veselý (1993)

Commentationes Mathematicae Universitatis Carolinae

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For convex continuous functions f , g defined respectively in neighborhoods of points x , y in a normed linear space, a formula for the distance between ∂ f ( x ) and ∂ g ( y ) in terms of f , g (i.eẇithout using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from [4], implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally...

Singular points of order k of Clarke regular and arbitrary functions

Luděk Zajíček (2012)

Commentationes Mathematicae Universitatis Carolinae

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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 d.c. C 1 function need not be difference of convex C 1 functions

David Pavlica (2005)

Commentationes Mathematicae Universitatis Carolinae

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In [2] a delta convex function on ℝ 2 is constructed which is strictly differentiable at 0 but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class C 1 ( ℝ 2 ) which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.