On some results about convex functions of order
M. Obradović, S. Owa (1986)
Matematički Vesnik
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M. Obradović, S. Owa (1986)
Matematički Vesnik
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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 is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.
Stefan Müller, Vladimír Šverák (1999)
Journal of the European Mathematical Society
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We study solutions of first order partial differential relations , where is a Lipschitz map and is a bounded set in matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of and second we replace Gromov’s −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...
Grzegorz Lewicki, Michael Prophet (2007)
Studia Mathematica
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We say that a function from 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”...
Witold Seredyński (2004)
Czechoslovak Mathematical Journal
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A closed convex set in a local convex topological Hausdorff spaces is called locally nonconical (LNC) if for every there exists an open neighbourhood of such that . A set is local cylindric (LC) if for , , there exists an open neighbourhood of such that (equivalently: ) is a union of open segments parallel to . In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication was proved in...
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), . A set S of vertices is a detour convex set if . The detour convex hull is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...
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 between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length 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 of a...
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 which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family involves the concept of cumulant transformation and a standard homogenization procedure.
Fethi Kadhi (2002)
RAIRO - Operations Research - Recherche Opérationnelle
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We investigate the minima of functionals of the form where is strictly convex. The admissible functions are not necessarily convex and satisfy on , , , is a fixed function on . We show that the minimum is attained by , the convex envelope of .
Katsuro Sakai, Zhongqiang Yang (2007)
Bulletin of the Polish Academy of Sciences. Mathematics
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Let be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that for every n > 1 whereas .
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 . 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 -spaces in for .
Prakash G. Umarani (1983)
Annales Polonici Mathematici
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