Estimates on inner and outer radii of unit balls in normed spaces

Horst Martini; Zokhrab Mustafaev

Displaying similar documents to “Estimates on inner and outer radii of unit balls in normed spaces”

Lineability and spaceability on vector-measure spaces

Giuseppina Barbieri, Francisco J. García-Pacheco, Daniele Puglisi (2013)

Studia Mathematica

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It is proved that if X is infinite-dimensional, then there exists an infinite-dimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that $c a (ℬ, λ, X) ∖ M_{σ}$ , the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear...

Continuous linear functionals on the space of Borel vector measures

Pola Siwek (2008)

Annales Polonici Mathematici

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We study properties of the space ℳ of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space ℳ is equipped with the Fortet-Mourier norm ${| | \cdot | |}_{ℱ}$ and the semivariation norm ||·||(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space $(ℳ, | | \cdot | |_{ℱ}) *$ is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals...

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

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Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a} (ξ) = [a / (h (ξ) + a)]$ , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain...

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^{M} = {[0, 1)}^{M}$ , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^{M}$ ; (3) the measures have the convolution property that $μ * L^{p} \subseteq L^{p + ε}$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $μ * L^{p} \subseteq L^{q}$ for any measure μ in our...

The multifractal box dimensions of typical measures

Frédéric Bayart (2012)

Fundamenta Mathematicae

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We compute the typical (in the sense of Baire’s category theorem) multifractal box dimensions of measures on a compact subset of $ℝ^{d}$ . Our results are new even in the context of box dimensions of measures.

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

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We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in $ℝ^{n_{i}}$ , i ≤ m, then for every F in the Grassmannian $G_{N, n}$ , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, $π_{F} (μ ₁ \otimes \dots \otimes μ ₘ)$ , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

A unified Lorenz-type approach to divergence and dependence

Teresa Kowalczyk

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AbstractThe paper deals with function-valued and numerical measures of absolute and directed divergence of one probability measure from another. In case of absolute divergence, some new results are added to the known ones to form a unified structure. In case of directed divergence, new concepts are introduced and investigated. It is shown that the notions of absolute and directed divergences complement each other and provide a good insight into the extent and the type of discrepancy...

Some singular measures on the circle which improve $L^{p}$ spaces

David L. Ritter (1987)

Colloquium Mathematicae

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Circumradius versus side lengths of triangles in linear normed spaces

Gennadiy Averkov (2007)

Colloquium Mathematicae

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Given a planar convex body B centered at the origin, we denote by ℳ ²(B) the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball B. For a triangle T in ℳ ²(B) we denote by $R_{B} (T)$ the least possible radius of a Minkowskian ball enclosing T. We remark that in the terminology of location science $R_{B} (T)$ is the optimum of the minimax location problem with distance induced by B and vertices of T as existing facilities (see, for instance, [HM03] and the references therein)....

Self-affine measures that are $L^{p}$ -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{q}$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^{p}$ -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$ -improving.

Estimates of capacity of self-similar measures

Jozef Myjak, Tomasz Szarek (2002)

Annales Polonici Mathematici

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We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems $(S_{i}, p_{i}) : i = 1, . . ., N$ where $S_{i}$ are bi-lipschitzean transformations.

Asymptotic behaviour of averages of k-dimensional marginals of measures on ℝⁿ

Jesús Bastero, Julio Bernués (2009)

Studia Mathematica

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We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set $x \in K : | P_{E} (x) | \leq t$ for a random k-dimensional subspace E ⊂ ℝⁿ and an isotropic convex body K ⊂ ℝⁿ. For k growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in $ℝ^{k}$ of a t-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on ℝⁿ.

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

Measures of pseudorandomness of finite binary lattices, I. The measures $Q_{k}$ , normality

Katalin Gyarmati, Christian Mauduit, András Sárközy (2010)

Acta Arithmetica

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Erratum to “Geometric rigidity of $\times m$ invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012))

Michael Hochman (2013)

Journal of the European Mathematical Society

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$L^{p}$ -improving properties of measures of positive energy dimension

Kathryn E. Hare, Maria Roginskaya (2005)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{p}$ to $L^{q}$ for some q > p. Positive measures which are $L^{p}$ -improving are known to have positive Hausdorff dimension. We extend this result to complex $L^{p}$ -improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of $L^{p}$ -functions.

Boundary value problem for an infinite system of second order differential equations in $ℓ_{p}$ spaces

Ishfaq Ahmad Malik, Tanweer Jalal (2020)

Mathematica Bohemica

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The concept of measures of noncompactness is applied to prove the existence of a solution for a boundary value problem for an infinite system of second order differential equations in $ℓ_{p}$ space. We change the boundary value problem into an equivalent system of infinite integral equations and result is obtained by using Darbo’s type fixed point theorem. The result is illustrated with help of an example.

Integral representation and relaxation for Junctionals defined on measures

Ennio De Giorgi, Luigi Ambrosio, Giuseppe Buttazzo (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Given a separable metric locally compact space $\Omega$ , and a positive finite non-atomic measure $\lambda$ on $\Omega$ , we study the integral representation on the space of measures with bounded variation $\Omega$ of the lower semicontinuous envelope of the functional $F(u)=\int_{\Omega}f(x,y)d\lambda\qquad u\in L^{1}(\Omega,\lambda,\mathbb{R}^{n})$ with respect to the weak convergence of measures.

Convex Corson compacta and Radon measures

Grzegorz Plebanek (2002)

Fundamenta Mathematicae

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Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $Σ (ℝ^{ω ₁})$ , and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{δ}$ -points.

Integral representation and relaxation for functionals defined on measures

Ennio De Giorgi, Luigi Ambrosio, Giuseppe Buttazzo (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Given a separable metric locally compact space $\Omega$ , and a positive finite non-atomic measure $\lambda$ on $\Omega$ , we study the integral representation on the space of measures with bounded variation $\Omega$ of the lower semicontinuous envelope of the functional $F(u)=\int_{\Omega}f(x,u)d\lambda\qquad u\in L^{1}(\Omega,\lambda,\mathbb{R}^{n})$ with respect to the weak convergence of measures.

Limit theorems for random fields

Nguyen van Thu

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CONTENTSIntroduction............................................................................................................................................................................ 51. Notation and preliminaries............................................................................................................................................ 52. Statement of the problem..................................................................................................................................................

On a decomposition of non-negative Radon measures

Bérenger Akon Kpata (2019)

Archivum Mathematicum

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We establish a decomposition of non-negative Radon measures on $ℝ^{d}$ which extends that obtained by Strichartz [6] in the setting of $α$ -dimensional measures. As consequences, we deduce some well-known properties concerning the density of non-negative Radon measures. Furthermore, some properties of non-negative Radon measures having their Riesz potential in a Lebesgue space are obtained.

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

Weak compactness in the space of operator valued measures $M_{b} a (Σ, (X, Y))$ and its applications

N.U. Ahmed (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{b a} (Σ, (X, Y))$ . This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{b a} (Σ, ₁ (X, Y))$ . This result has interesting applications in optimization and control theory as illustrated by several examples.

A theorem on isotropic spaces

Félix Cabello Sánchez (1999)

Studia Mathematica

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Let X be a normed space and $G_{F} (X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_{F} (X)$ acts transitively on the unit sphere then X must be an inner product space.

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

Maximal function and Carleson measures in the theory of Békollé-Bonami weights

Carnot D. Kenfack, Benoît F. Sehba (2016)

Colloquium Mathematicae

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Let ω be a Békollé-Bonami weight. We give a complete characterization of the positive measures μ such that $\int_{} | M_{ω} {f (z) |}^{q} d μ (z) \leq C (\int_{} {| f (z) |}^{p} {ω (z) d V (z))}^{q / p}$ and $μ (z \in : M f (z) > λ) \leq C / (λ^{q}) (\int_{} | f (z) {|^{p} ω (z) d V (z))}^{q / p}$ , where $M_{ω}$ is the weighted Hardy-Littlewood maximal function on the upper half-plane and 1 ≤ p,q <; ∞.

Some properties and applications of equicompact sets of operators

E. Serrano, C. Piñeiro, J. M. Delgado (2007)

Studia Mathematica

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Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence $(x_{k (n)}) ₙ$ such that $(T x_{k (n)}) ₙ$ is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness...

Level by level equivalence and the number of normal measures over $P_{κ} (λ)$

Arthur W. Apter (2007)

Fundamenta Mathematicae

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We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures $P_{κ} (λ)$ carries. In the first of these models, $P_{κ} (λ)$ carries $2^{2^{{[λ]}^{< κ}}}$ many normal measures, the maximal number. In the second of these models, $P_{κ} (λ)$ carries $2^{2^{{[λ]}^{< κ}}}$ many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and...

Path functionals over Wasserstein spaces

Alessio Brancolini, Giuseppe Buttazzo, Filippo Santambrogio (2006)

Journal of the European Mathematical Society

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Given a metric space $X$ we consider a general class of functionals which measure the cost of a path in $X$ joining two given points $x_{0}$ and $x_{1}$ , providing abstract existence results for optimal paths. The results are then applied to the case when $X$ is aWasserstein space of probabilities on a given set $Ω$ and the cost of a path depends on the value of classical functionals over measures. Conditions for linking arbitrary extremal measures $μ_{0}$ and $μ_{1}$ by means of finite cost paths are given. ...