Norms with locally Lipschitzian derivatives

Fabian, Marián; Whitfield, J. H.; Zizler, Václav

doi:10.1007/bf02760975

Cited by 48 publications

(42 citation statements)

References 16 publications

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“…The following result extends what formerly was known for C 2 -smooth LUR norms (see, e.g., [FHHMZ,Exercise 9.16]) and later for C 2 -smooth norms with a strongly exposed point on its unit sphere [FWZ83,Theorem 3.3]. Proof.…”

Section: Theorem 1 Let (X ∥ · ∥ 0 ) Be a Banach Space Having A Countsupporting

confidence: 80%

A note on extreme points of $C^\infty $-smooth balls in polyhedral spaces

Guirao¹,

Montesinos²,

Zizler

2015

Proc. Amer. Math. Soc.

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Morris [Mo83] proved that every separable Banach space X that contains an isomorphic copy of c0 has an equivalent strictly convex norm such that all points of its unit sphere SX are unpreserved extreme, i.e., they are no longer extreme points of BX * * . We use a result of Hájek [Ha95] to prove that any separable infinite-dimensional polyhedral Banach space has an equivalent C ∞ -smooth and strictly convex norm with the same property as in Morris' result. We additionally show that no point on the sphere of a C 2 -smooth equivalent norm on a polyhedral infinite-dimensional space can be strongly extreme, i.e., there is no point x on the sphere for which a sequence (hn) in X with ∥hn∥ ̸ → 0 exists such that ∥x ± hn∥ → 1.

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Section: Theorem 1 Let (X ∥ · ∥ 0 ) Be a Banach Space Having A Countsupporting

confidence: 80%

A note on extreme points of $C^\infty $-smooth balls in polyhedral spaces

Guirao¹,

Montesinos²,

Zizler

2015

Proc. Amer. Math. Soc.

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“…When X has a power modulus of smoothness [20] one can be more specific. By extending the argument in [15,Lemma 2.4], one may show that X has a power modulus of smoothness £s (1 < s < 2) if and only if the function ga :-|| • ||s/s satisfies (2)(3)(4) \\Vga(x)-\7gs(y)\\<C\\x-y\r1 for x and y in X, where C is independent of x and y. It follows that As will also satisfy (2.4) which is to say that As has a Holder-continuous derivative.…”

Section: Introductionmentioning

confidence: 93%

A Smooth Variational Principle With Applications to Subdifferentiability and to Differentiability of Convex Functions

Borwein¹,

Preiss²

1987

Transactions of the American Mathematical Society

104

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ABSTRACT. We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Fréchet) smooth renorm is densely Gâteaux (weak Hadamard, Fréchet) differentiable. Our technique relies on a more powerful analogue of Ekeland's variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.

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“…Speaking about higher order differentiability of norms, it was shown in [2] that a Banach space X is isomorphic to a Hilbert space if on both X and X* there exist real valued functions ~ with bounded nonempty support and such that ~o' are locally Lipschitzian. It also is known that a Banach space X which admits a real valued twice continuously differentiable function ~o with bounded nonempty support is isomorphic to a FIilbert space provided any infinite dimensional subspace of X contains an infinite dimensional subspace which is isomorphic to a Hilbert space [9].…”

Section: A Note On Cotype Of Smooth Spacesmentioning

confidence: 99%

A note on cotype of smooth spaces

Deville

Zizler

1988

Manuscripta Math

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Abstract. For sufficiently smooth Banach spaces weak cotype 2 implies Hilbert space.It is well known that differentiability of the norm on a given Banach space X has a significant impact on the structure of X. Speaking about higher order differentiability of norms, it was shown in [2] that a Banach space X is isomorphic to a Hilbert space if on both X and X* there exist real valued functions ~ with bounded nonempty support and such that ~o' are locally Lipschitzian. It also is known that a Banach space X which admits a real valued twice continuously differentiable function ~o with bounded nonempty support is isomorphic to a FIilbert space provided any infinite dimensional subspace of X contains an infinite dimensional subspace which is isomorphic to a Hilbert space [9]. There are many spaces which do not admit real valued twice differentiable functions with bounded support and yet have norms with differential which is Lipschitzian on the sphere [12], I4],

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Norms with locally Lipschitzian derivatives

Cited by 48 publications

References 16 publications

A note on extreme points of $C^\infty $-smooth balls in polyhedral spaces

A note on extreme points of $C^\infty $-smooth balls in polyhedral spaces

A Smooth Variational Principle With Applications to Subdifferentiability and to Differentiability of Convex Functions

A note on cotype of smooth spaces

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