Cardinality of some convex sets and of their sets of extreme points

Zbigniew Lipecki

Displaying similar documents to “Cardinality of some convex sets and of their sets of extreme points”

On co-ordinated quasi-convex functions

M. Emin Özdemir, Ahmet Ocak Akdemir, Çetin Yıldız (2012)

Czechoslovak Mathematical Journal

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A function $f : I \to ℝ$ , where $I \subseteq ℝ$ is an interval, is said to be a convex function on $I$ if $f (t x + (1 - t) y) \leq t f (x) + (1 - t) f (y)$ holds for all $x, y \in I$ and $t \in [0, 1]$ . There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations. We define some new classes of convex functions that we name quasi-convex, Jensen-convex, Wright-convex, Jensen-quasi-convex and Wright-quasi-convex...

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 $f$ .

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...

Compactness and extreme points of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki

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The memoir is based on a series of six papers by the author published over the years 1995-2007. It continues the work of D. Plachky (1970, 1976). It also owes some inspiration, among others, to papers by J. Łoś and E. Marczewski (1949), D. Bierlein and W. J. A. Stich (1989), D. Bogner and R. Denk (1994), and A. Ülger (1996). Let and ℜ be algebras of subsets of a set Ω with ⊂ ℜ. Given a quasi-measure μ on , i.e., μ ∈ ba₊(), we denote by E(μ) the convex set of all quasi-measure extensions...

$ℒ_{p}$ -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.

Measure and Helly's Intersection Theorem for Convex Sets

N. Stavrakas (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $ℱ = F_{α}$ be a uniformly bounded collection of compact convex sets in ℝ ⁿ. Katchalski extended Helly’s theorem by proving for finite ℱ that dim (⋂ ℱ) ≥ d, 0 ≤ d ≤ n, if and only if the intersection of any f(n,d) elements has dimension at least d where f(n,0) = n+1 = f(n,n) and f(n,d) = maxn+1,2n-2d+2 for 1 ≤ d ≤ n-1. An equivalent statement of Katchalski’s result for finite ℱ is that there exists δ > 0 such that the intersection of any f(n,d) elements of ℱ contains a d-dimensional ball...

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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Unique $a$ -closure for some $ℓ$ -groups of rational valued functions

Anthony W. Hager, Chawne M. Kimber, Warren W. McGovern (2005)

Czechoslovak Mathematical Journal

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Usually, an abelian $ℓ$ -group, even an archimedean $ℓ$ -group, has a relatively large infinity of distinct $a$ -closures. Here, we find a reasonably large class with unique and perfectly describable $a$ -closure, the class of archimedean $ℓ$ -groups with weak unit which are “ $ℚ$ -convex”. ( $ℚ$ is the group of rationals.) Any $C (X, ℚ)$ is $ℚ$ -convex and its unique $a$ -closure is the Alexandroff algebra of functions on $X$ defined from the clopen sets; this is sometimes $C (X)$ .

Continuous version of the Choquet integral representation theorem

Piotr Puchała (2005)

Studia Mathematica

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Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if $s u p_{| | f | | \leq 1} | f (x) - \int_{K} f d μ | < γ$ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family $(μ_{t})$ of...

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.

An intersection theorem for set-valued mappings

Ravi P. Agarwal, Mircea Balaj, Donal O'Regan (2013)

Applications of Mathematics

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Given a nonempty convex set $X$ in a locally convex Hausdorff topological vector space, a nonempty set $Y$ and two set-valued mappings $T : X ⇉ X$ , $S : Y ⇉ X$ we prove that under suitable conditions one can find an $x \in X$ which is simultaneously a fixed point for $T$ and a common point for the family of values of $S$ . Applying our intersection theorem we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems. ...

Graphs with convex domination number close to their order

Joanna Cyman, Magdalena Lemańska, Joanna Raczek (2006)

Discussiones Mathematicae Graph Theory

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For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance $d_{G} (u, v)$ between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length $d_{G} (u, v)$ is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number $γ_{c o n} (G)$ of a...