Topological classification of closed convex sets in Fréchet spaces

Taras Banakh; Robert Cauty

Displaying similar documents to “Topological classification of closed convex sets in Fréchet spaces”

On Fréchet differentiability of convex functions on Banach spaces

Wee-Kee Tang (1995)

Commentationes Mathematicae Universitatis Carolinae

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Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function $f$ defined on a separable Banach space are studied. The conditions are in terms of a majorization of $f$ by a $C^{1}$ -smooth function, separability of the boundary for $f$ or an approximation of $f$ by Fréchet smooth convex functions.

On closed sets with convex projections in Hilbert space

Stoyu Barov, Jan J. Dijkstra (2007)

Fundamenta Mathematicae

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Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set $^{k} (B)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that $^{k} (B)$ is precisely the intersection of...

Generalized characterization of the convex envelope of a function

Fethi Kadhi (2002)

RAIRO - Operations Research - Recherche Opérationnelle

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We investigate the minima of functionals of the form $\int_{[a, b]} g (\dot{u} (s)) d s$ where $g$ is strictly convex. The admissible functions $u : [a, b] \to ℝ$ are not necessarily convex and satisfy $u \leq f$ on $[a, b]$ , $u (a) = f (a)$ , $u (b) = f (b)$ , $f$ is a fixed function on $[a, b]$ . We show that the minimum is attained by $\bar{f}$ , the convex envelope of $f$ .

Topological classification of strong duals to nuclear (LF)-spaces

Taras Banakh (2000)

Studia Mathematica

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We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: $ℝ^{ω}$ , $ℝ^{\infty}$ , $Q \times ℝ^{\infty}$ , $ℝ^{ω} \times ℝ^{\infty}$ , or ${(ℝ^{\infty})}^{ω}$ , where $ℝ^{\infty} = l i m ℝ^{n}$ and $Q = {[- 1, 1]}^{ω}$ . In particular, the Schwartz space D’ of distributions is homeomorphic to ${(ℝ^{\infty})}^{ω}$ . As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to $ℝ^{\infty}$ or to $Q \times ℝ^{\infty}$ . In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic...

A hereditary property in locally convex spaces

Manuel Valdivia (1971)

Annales de l'institut Fourier

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If $E$ is an infrabarrelled space, it is proved that any subspace $F$ , of finite codimension, is infrabarrelled.

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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Countably convex $G_{δ}$ sets

Vladimir Fonf, Menachem Kojman (2001)

Fundamenta Mathematicae

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We investigate countably convex $G_{δ}$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets...

Convex universal fixers

Magdalena Lemańska, Rita Zuazua (2012)

Discussiones Mathematicae Graph Theory

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In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and $E (π G) = E (G) \cup E (G^{'}) \cup M_{π}$ , where $M_{π} = u π (u) | u \in V (G)$ . Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept...

A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces

Taras Banakh, Ivan Hetman (2012)

Studia Mathematica

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A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ⊂ X at Hausdorff distance < ε from C. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space $C o n v_{} (X)$ of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a...

$ℒ_{p}$ -Spaces and injective locally convex spaces

Paweł Domański

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1985 Mathematics Subject Classification: Primary 46A05, 46A06, 46A12, 46A20, 46A22, 46A99, 46M10, 03C20; Secondary 46E10, 46E15, 46E30, 46C99.

The Young inequality and the Δ₂-condition

Philippe Laurençot (2002)

Colloquium Mathematicae

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If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality $x y \leq ε φ (x) + C_{ε} φ * (y)$ is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.

Smoothing a polyhedral convex function via cumulant transformation and homogenization

Alberto Seeger (1997)

Annales Polonici Mathematici

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Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family ${g ₜ}_{t > 0}$ which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family ${g ₜ}_{t > 0}$ involves the concept of cumulant transformation and a standard homogenization procedure.

Topological Structure of Non-separable Sigma-locally Compact Convex Sets

Iryna Banakh, Taras Banakh, Katsuhisa Koshino (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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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.