Convex universal fixers

Magdalena Lemańska; Rita Zuazua

Displaying similar documents to “Convex universal fixers”

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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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...

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...

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.

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 vertex detour hull number of a graph

A.P. Santhakumaran, S.V. Ullas Chandran (2012)

Discussiones Mathematicae Graph Theory

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For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_{D} [S] = ⋃_{x, y \in S} I_{D} [x, y]$ . A set S of vertices is a detour convex set if $I_{D} [S] = S$ . The detour convex hull ${[S]}_{D}$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...

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...

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”...

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 $C o n v_{F} (ℝ) \approx ℝ \times$ .

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

B. Mirković (1970)

Matematički Vesnik

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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.

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 $f$ .

On the ψ₂-behaviour of linear functionals on isotropic convex bodies

G. Paouris (2005)

Studia Mathematica

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The slicing problem can be reduced to the study of isotropic convex bodies K with $d i a m (K) \leq c \sqrt n L_{K}$ , where $L_{K}$ is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that ${| | ⟨ \cdot, θ ⟩ | |}_{ψ ₂} \leq C L_{K}$ for all θ in a subset U of $S^{n - 1}$ with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that $m a x_{θ \in S^{n - 1}} {| | ⟨ \cdot, θ ⟩ | |}_{ψ ₂} \geq c ∜ n L_{K}$ . In a different direction, we show that good average ψ₂-behaviour of linear functionals...

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

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

Matematički Vesnik

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The forcing convexity number of a graph

Gary Chartrand, Ping Zhang (2001)

Czechoslovak Mathematical Journal

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For two vertices $u$ and $v$ of a connected graph $G$ , the set $I (u, v)$ consists of all those vertices lying on a $u$ – $v$ geodesic in $G$ . For a set $S$ of vertices of $G$ , the union of all sets $I (u, v)$ for $u, v \in S$ is denoted by $I (S)$ . A set $S$ is a convex set if $I (S) = S$ . The convexity number $c o n (G)$ of $G$ is the maximum cardinality of a proper convex set of $G$ . A convex set $S$ in $G$ with $| S | = c o n (G)$ is called a maximum convex set. A subset $T$ of a maximum convex set $S$ of a connected graph $G$ is called a forcing subset for $S$ if $S$ is the unique maximum...