Displaying similar documents to “Morse-Sard theorem for delta-convex curves”

When ( E , σ ( E , E ' ) ) is a D F -space?

Dorota Krassowska, Wiesƚaw Śliwa (1992)

Commentationes Mathematicae Universitatis Carolinae

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Let ( E , t ) be a Hausdorff locally convex space. Either ( E , σ ( E , E ' ) ) or ( E ' , σ ( E ' , E ) ) is a D F -space iff E is of finite dimension (THEOREM). This is the most general solution of the problem studied by Iyahen [2] and Radenovič [3].

Some characterization of locally nonconical convex sets

Witold Seredyński (2004)

Czechoslovak Mathematical Journal

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A closed convex set Q in a local convex topological Hausdorff spaces X is called locally nonconical (LNC) if for every x , y ∈ Q there exists an open neighbourhood U of x such that ( U ∩ Q ) + 1 2 ( y - x ) ⊂ Q . A set Q is local cylindric (LC) if for x , y ∈ Q , x ≠ y , z ∈ ( x , y ) there exists an open neighbourhood U of z such that U ∩ Q (equivalently: b d ( Q ) ∩ U ) is a union of open segments parallel to [ x , y ] . In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication L N C ⇒ L C was proved in...

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations D u ∈ K , where u : Ω ⊂ ℝ n → ℝ m is a Lipschitz map and K is a bounded set in m × n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of D u and second we replace Gromov’s P −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

Separated sequences in uniformly convex Banach spaces

J. M. A. M. van Neerven (2005)

Colloquium Mathematicae

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We give a characterization of uniformly convex Banach spaces in terms of a uniform version of the Kadec-Klee property. As an application we prove that if (xₙ) is a bounded sequence in a uniformly convex Banach space X which is ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence ( x n j ) of (xₙ) such that i n f j ≠ k | | x - ( x n j - x n k ) | | ≥ 1 + δ X ( 2 / 3 ε ) , where δ X is the modulus of convexity of X. From this we deduce that the unit sphere of every infinite-dimensional uniformly convex Banach space...

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...

Fixed points of asymptotically regular mappings in spaces with uniformly normal structure

Jarosław Górnicki (1991)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that: for every Banach space X which has uniformly normal structure there exists a k > 1 with the property: if A is a nonempty bounded closed convex subset of X and T : A → A is an asymptotically regular mapping such that lim inf n → ∞ | | | T n | | | < k , where | | | T | | | is the Lipschitz constant (norm) of T , then T has a fixed point in A .