The radius of close-to-convexity of $V_k(ϱ)$

Prakash G. Umarani

Displaying similar documents to “The radius of close-to-convexity of $V_{k} (ϱ)$ ”

Remarks on Convexity in Dimension (2,2)

Wojciech Kryński (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider different convexity notions for functions $F : ℝ^{2 \times 2} \to ℝ$ . We give a new characterisation of polyconvexity and a sufficient condition for quasiconvexity.

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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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.

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

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

Matematički Vesnik

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A characterization of sets in $ℝ^{2}$ with DC distance function

Dušan Pokorný, Luděk Zajíček (2022)

Czechoslovak Mathematical Journal

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We give a complete characterization of closed sets $F \subset ℝ^{2}$ whose distance function $d_{F} : = dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $ℝ^{2}$ ). Using this characterization, a number of properties of such sets is proved.

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 $C o n v_{F} (ℝ) \approx ℝ \times$ .

Product property for capacities in $ℂ^{N}$

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: $C_{ν} (E ₁ \times E ₂) = m i n (C_{ν ₁} (E ₁), C_{ν ₂} (E ₂))$ , where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N ₁ + N ₂}$ , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$ , denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

Operations between sets in geometry

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 $n$ -dimensional Euclidean space $ℝ^{n}$ . It is proved that if $n \geq 2$ , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $G L (n)$ covariant, and associative if and only if it is $L_{p}$ addition for some $1 \leq p \leq \infty$ . It is also demonstrated...

An integral operator on the classes $𝒮^{*} (α)$ and $𝒞𝒱ℋ (β)$

Nicoleta Ularu, Nicoleta Breaz (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $α$ and from the class $𝒞𝒱ℋ (β)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $𝒞𝒱ℋ (β)$ .

Łojasiewicz-Siciak condition for the pluricomplex Green function

Marta Kosek (2011)

Banach Center Publications

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A compact set $K \subset ℂ^{N}$ satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that $V_{K} (z) \geq B {(d i s t (z, K))}^{β}$ if dist(z,K) ≤ 1. (LS) Here $V_{K}$ 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).

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

B. Mirković (1970)

Matematički Vesnik

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Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

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If $E = e_{i}$ and $F = f_{i}$ 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 $a_{i, j}$ with norm $| | a_{i, j} {| |}_{E (F)} = | | \sum_{k} | | \sum_{l} a_{k, l} f_{l} | | e_{k} | |$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

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...

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 $0 < p < 1$ .

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

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We consider the integral functional $J (u) = \int_{Ω} [f (| D u |) - u] d x$ , $u \in W_{0}^{1, 1} (Ω)$ , where $Ω \subset ℝ^{n}$ , $n \geq 2$ , is a nonempty bounded connected open subset of $ℝ^{n}$ with smooth boundary, and $ℝ ∋ s \mapsto f (| s |)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball.

On starshapedness of the union of closed sets in $R^{n}$

Krzysztof Kołodziejczyk (1987)

Colloquium Mathematicae

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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”...

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...

A characterization of the disc $D^{n}$

Th. Friedrich (1974)

Colloquium Mathematicae

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Quantitative stability for sumsets in $ℝ^{n}$

Alessio Figalli, David Jerison (2015)

Journal of the European Mathematical Society

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Given a measurable set $A \subset ℝ^{n}$ of positive measure, it is not difficult to show that $| A + A | = | 2 A |$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(| A + A | - | 2 A |) / | A |$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(| A + A | - | 2 A |) / | A |$ .

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class $ℋ_{0}$ of sense-preserving harmonic functions $f = h + \bar{g}$ defined in the unit disk $| z | < 1$ and normalized so that $h (0) = 0 = h^{'} (0) - 1$ and $g (0) = 0 = g^{'} (0)$ , where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $𝒫_{H}^{0} (α)$ and $𝒢_{H}^{0} (β)$ of functions from $ℋ_{0}$ and show that if $f \in 𝒫_{H}^{0} (α)$ and $F \in 𝒢_{H}^{0} (β)$ , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $α$ and $β$ are satisfied. In the second part we study the harmonic sections...

Spectral radius of operators associated with dynamical systems in the spaces C(X)

Krzysztof Zajkowski (2005)

Banach Center Publications

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We consider operators acting in the space C(X) (X is a compact topological space) of the form $A u (x) = (\sum_{k = 1}^{N} e^{φ_{k}} T_{α_{k}}) u (x) = \sum_{k = 1}^{N} e^{φ_{k} (x)} u (α_{k} (x))$ , u ∈ C(X), where $φ_{k} \in C (X)$ and $α_{k} : X \to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $φ = {(φ_{k})}_{k = 1}^{N}$ . We prove that $l n (r (A)) = λ (φ) = m a x_{ν \in M e s} \sum_{k = 1}^{N} \int_{X} φ_{k} d ν_{k} - λ * (ν)$ , where Mes is the set of all probability vectors of measures $ν = {(ν_{k})}_{k = 1}^{N}$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...

On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

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On the Schröder equation

M. Kuczma

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CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. $C^{p}$ solutions in $ℭ$ *................................................ 74. Change of variables..............................................................................

Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi (2015)

Journal of the European Mathematical Society

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We consider the functional $ℐ_{Ω} (v) = \int_{Ω} [f (| D v |) - v] d x,$ where $Ω$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$ , Crasta [Cr1] has shown that if $ℐ_{Ω}$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball. With some restrictions on $f$ , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss...

Displaying similar documents to “The radius of close-to-convexity of V k ( ϱ ) ”

Displaying similar documents to “The radius of close-to-convexity of $V_{k} (ϱ)$ ”