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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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 .
Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)
Journal of the European Mathematical Society
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An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in -dimensional Euclidean space . It is proved that if , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, covariant, and associative if and only if it is addition for some . It is also demonstrated...
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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 .
G. Paouris (2005)
Studia Mathematica
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The slicing problem can be reduced to the study of isotropic convex bodies K with , where is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that for all θ in a subset U of 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 . In a different direction, we show that good average ψ₂-behaviour of linear functionals...
Mirosław Baran, Leokadia Bialas-Ciez (2012)
Annales Polonici Mathematici
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The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: , where and are respectively a compact set and a norm in (j = 1,2), and ν is a norm in , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of , denote by C(E) the standard L-capacity and by the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...
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.
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”...
B. Mirković (1970)
Matematički Vesnik
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Alessio Figalli, David Jerison (2015)
Journal of the European Mathematical Society
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Given a measurable set of positive measure, it is not difficult to show that if and only if is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If is small, is close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between and its convex hull in terms of .
Edvard Kramar (2016)
Commentationes Mathematicae Universitatis Carolinae
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Denote by the commutator of two bounded operators and acting on a locally convex topological vector space. If , we show that is a quasinilpotent operator and we prove that if is a compact operator, then is a Riesz operator.
Liulan Li, Saminathan Ponnusamy (2016)
Czechoslovak Mathematical Journal
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We consider the class of sense-preserving harmonic functions defined in the unit disk and normalized so that and , where and are analytic in the unit disk. In the first part of the article we present two classes and of functions from and show that if and , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters and are satisfied. In the second part we study the harmonic sections...
M. Kuczma
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CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. solutions in *................................................ 74. Change of variables..............................................................................
Gideon Schechtman (2013)
Studia Mathematica
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If and are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices with norm embeds into L₁. This generalizes a recent result of Prochno and Schütt.
Trung Hoa Dinh, Sima Ahsani, Tin-Yau Tam (2016)
Czechoslovak Mathematical Journal
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We study some geometric properties associated with the -geometric means of two positive definite matrices and . Some geodesical convexity results with respect to the Riemannian structure of the positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding...
Graziano Crasta (2006)
Journal of the European Mathematical Society
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We consider the integral functional , , where , , is a nonempty bounded connected open subset of with smooth boundary, and is a convex, differentiable function. We prove that if admits a minimizer in depending only on the distance from the boundary of , then must be a ball.
Marta Kosek (2011)
Banach Center Publications
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A compact set satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that if dist(z,K) ≤ 1. (LS) Here denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).
Agnieszka Bogdewicz, Jerzy Grzybowski (2009)
Banach Center Publications
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Let be a Minkowski space with a unit ball and let be the Hausdorff metric induced by 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 for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace over any two-dimensional Minkowski space.
S. Owa, C. Y. Shen (1988)
Matematički Vesnik
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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...
Chi-Kwong Li, Yiu-Tung Poon (2009)
Studia Mathematica
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Let W(A) and be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, can be obtained as the intersection of all sets of the form , where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in...