On convex and *-concave multifunctions

Bożena Piątek

Displaying similar documents to “On convex and *-concave multifunctions”

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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Minimax theorems without changeless proportion

Liang-Ju Chu, Chi-Nan Tsai (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: $i n f_{y \in Y} s u p_{x \in X} f (x, y) \leq s u p_{x \in X} i n f_{y \in Y} g (x, y)$ . We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $s u p_{x \in X} f (x, y) \leq s u p_{x \in X} g (x, y)$ , ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some...

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.

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 $D u \in K$ , where $u : Ω \subset ℝ^{n} \to ℝ^{m}$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ 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...

Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment

Agnieszka Bogdewicz, Jerzy Grzybowski (2009)

Banach Center Publications

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Let $(ℝ, | | \cdot | |_{)}$ be a Minkowski space with a unit ball and let $ϱ_{H}^{}$ be the Hausdorff metric induced by ${| | \cdot | |}_{}$ in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to $ϱ_{H}^{B ⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(², ϱ_{H}^{})$ over any two-dimensional Minkowski 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...

Poincaré Inequalities and Moment Maps

Bo’az Klartag (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in $ℝ^{n}$ . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of $ℓ_{p}$ -spaces in $ℝ^{n}$ for $0 < p < 1$ .

The radius of close-to-convexity of $V_{k} (ϱ)$

Prakash G. Umarani (1983)

Annales Polonici Mathematici

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Minimal multi-convex projections

Grzegorz Lewicki, Michael Prophet (2007)

Studia Mathematica

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We say that a function from $X = C^{L} [0, 1]$ is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape”...

CM-Selectors for pairs of oppositely semicontinuous multivalued maps with $_{p}$ -decomposable values

Hôǹg Thái Nguyêñ, Maciej Juniewicz, Jolanta Ziemińska (2001)

Studia Mathematica

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We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_{p} (T, E)$ -decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_{p} (T, E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x)...

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 $f$ .

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Katsuro Sakai, Zhongqiang Yang (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $C o n v_{F} (ℝ ⁿ)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $C o n v_{F} (ℝ ⁿ) \approx ℝ ⁿ \times Q$ for every n > 1 whereas $C o n v_{F} (ℝ) \approx ℝ \times$ .

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.

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 $x, y \in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U \cap Q) + \frac{1}{2} (y - x) \subset Q$ . A set $Q$ is local cylindric (LC) if for $x, y \in Q$ , $x \neq y$ , $z \in (x, y)$ there exists an open neighbourhood $U$ of $z$ such that $U \cap Q$ (equivalently: $b d (Q) \cap U$ ) 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 $L N C \Rightarrow L C$ was proved in...

A note on the functions $(S_{n} f) (z)$ defined by Ruscheweyn

S. Owa, C. Y. Shen (1988)

Matematički Vesnik

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Differentiation of n-convex functions

H. Fejzić, R. E. Svetic, C. E. Weil (2010)

Fundamenta Mathematicae

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The main result of this paper is that if f is n-convex on a measurable subset E of ℝ, then f is n-2 times differentiable, n-2 times Peano differentiable and the corresponding derivatives are equal, and $f^{(n - 1)} = f_{(n - 1)}$ except on a countable set. Moreover $f_{(n - 1)}$ is approximately differentiable with approximate derivative equal to the nth approximate Peano derivative of f almost everywhere.

A note on locally convex spaces with a basic sequence of $β$ -disks

B. Mirković (1970)

Matematički Vesnik

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Some approximate fixed point theorems without continuity of the operator using auxiliary functions

Sumit Chandok, Arslan Hojjat Ansari, Tulsi Dass Narang (2019)

Mathematica Bohemica

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We introduce partial generalized convex contractions of order $4$ and rank $4$ using some auxiliary functions. We present some results on approximate fixed points and fixed points for such class of mappings having no continuity condition in $α$ -complete metric spaces and $μ$ -complete metric spaces. Also, as an application, some fixed point results in a metric space endowed with a binary relation and some approximate fixed point results in a metric space endowed with a graph have been obtained....

Lower semicontinuous envelopes in $W^{1, 1} \times L^{p}$

Ana Margarida Ribeiro, Elvira Zappale (2014)

Banach Center Publications

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The lower semicontinuity of functionals of the type $\int_{Ω} f (x, u, v, \nabla u) d x$ with respect to the $(W^{1, 1} \times L^{p})$ -weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1, 1} \times L^{p}$ for the lower semicontinuous envelope is also provided.