Generalized characterization of the convex envelope of a function

Fethi Kadhi

Displaying similar documents to “Generalized characterization of the convex envelope of a function”

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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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...

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.

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 $0 < p < 1$ .

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...

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”...

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...

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

Prakash G. Umarani (1983)

Annales Polonici Mathematici

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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...

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...

Convexity of sublevel sets of plurisubharmonic extremal functions

Finnur Lárusson, Patrice Lassere, Ragnar Sigurdsson (1998)

Annales Polonici Mathematici

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Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function $u_{E, X}$ for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of $u_{E, X}$ are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

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 $C o n v_{F} (ℝ) \approx ℝ \times$ .

On convex and *-concave multifunctions

Bożena Piątek (2005)

Annales Polonici Mathematici

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A continuous multifunction F:[a,b] → clb(Y) is *-concave if and only if the inclusion $1 / (t - s) \int_{s}^{t} F (x) d x \subset (F (s) \frac{*}{+} F (t)) / 2$ holds for every s,t ∈ [a,b], s < t.

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 $i n f_{j \neq k} | | x - (x_{n_{j}} - x_{n_{k}}) | | \geq 1 + δ_{X} (2 / 3 ε)$ , 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...

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

B. Mirković (1970)

Matematički Vesnik

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