scholarly journals Compactness of a conformal boundary of the Euclidean unit ball

2011 ◽  
Vol 36 ◽  
pp. 3-20
Author(s):  
Päivi Lammi
Keyword(s):  
Unit Ball ◽  
Conformal Boundary ◽  
Euclidean Unit Ball

scholarly journals A Note on Strongly Starlike Mappings in Several Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Hidetaka Hamada ◽  
Tatsuhiro Honda ◽  
Gabriela Kohr ◽  
Kwang Ho Shon
Keyword(s):  
Unit Ball ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Biholomorphic Mapping ◽  
Strongly Starlike ◽  
Starlike Mappings ◽  
Euclidean Unit Ball

Letfbe a normalized biholomorphic mapping on the Euclidean unit ball𝔹ninℂnand letα∈0,1. In this paper, we will show that iffis strongly starlike of orderαin the sense of Liczberski and Starkov, then it is also strongly starlike of orderαin the sense of Kohr and Liczberski. We also give an example which shows that the converse of the above result does not hold in dimensionn≥2.

Parametric Representation of Univalent Mappings in Several Complex Variables

2002 ◽  
Vol 54 (2) ◽  
pp. 324-351 ◽  
Author(s):  
Ian Graham ◽  
Hidetaka Hamada ◽  
Gabriela Kohr
Keyword(s):  
Unit Ball ◽  
Existence Theorems ◽  
Parametric Representation ◽  
Positive Real Part ◽  
Complex Variables ◽  
Holomorphic Mappings ◽  
Several Complex Variables ◽  
Positive Real ◽  
Loewner Equation ◽  
Euclidean Unit Ball

AbstractLet B be the unit ball of with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S0(B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of S0(B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S0(B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree

2021 ◽  
Vol 127 (2) ◽  
pp. 337-360
Author(s):  
Norman Levenberg ◽  
Franck Wielonsky
Keyword(s):  
Convex Body ◽  
Unit Ball ◽  
General Formula ◽  
Plurisubharmonic Function ◽  
Integral Formula ◽  
Compact Set ◽  
Transfinite Diameter ◽  
Robin Function ◽  
Euclidean Unit Ball ◽  
Definition Of

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

Convex Subordination Chains in Several Complex Variables

2009 ◽  
Vol 61 (3) ◽  
pp. 566-582 ◽  
Author(s):  
Ian Graham ◽  
Hidetaka Hamada ◽  
Gabriela Kohr ◽  
John A. Pfaltzgraff
Keyword(s):  
Unit Ball ◽  
Sufficient Conditions ◽  
Complex Variables ◽  
Necessary And Sufficient Conditions ◽  
Several Complex Variables ◽  
Sufficient Condition ◽  
Subordination Chain ◽  
Euclidean Unit Ball ◽  
Necessary And Sufficient

Abstract.In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ℂn. We also obtain a sufficient condition for injectivity off(z/‖z‖, ‖z‖) onBn\ ﹛0﹜, wheref(z,t) is a convex subordination chain over (0, 1).

scholarly journals A note on norms of signed sums of vectors

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Giorgos Chasapis ◽  
Nikos Skarmogiannis
Keyword(s):  
Convex Body ◽  
Unit Ball ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Euclidean Unit Ball ◽  
Arbitrary Norm

AbstractImproving a result of Hajela, we show for every function f with limn→∞f(n) = ∞ and f(n) = o(n) that there exists n0 = n0(f) such that for every n ⩾ n0 and any S ⊆ {–1, 1}n with cardinality |S| ⩽ 2n/f(n) one can find orthonormal vectors x1, …, xn ∈ ℝn satisfying $\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all (ε1, …, εn) ∈ S. We obtain analogous results in the case where x1, …, xn are independent random points uniformly distributed in the Euclidean unit ball $\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the $\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$-norm is replaced by an arbitrary norm on ℝn.

scholarly journals Almost contractive retractions in Orlicz spaces

2003 ◽  
Vol 68 (3) ◽  
pp. 353-369 ◽  
Author(s):  
Grzegorz Lewicki ◽  
Giulio Trombetta
Keyword(s):  
Finite Elements ◽  
Convex Function ◽  
Unit Ball ◽  
Measure Of Noncompactness ◽  
Orlicz Spaces ◽  
Closed Unit Ball ◽  
Hausdorff Measure Of Noncompactness ◽  
Contraction Theorem ◽  
Euclidean Unit Ball ◽  
Dimensional Lebesgue Measure

Let Bk denote the Euclidean unit ball in ℝκ equipped with the k-dimensional Lebesgue measure and let φ: ℝ+ → ℝ+ be a convex function satisfying φ(0) = 0, φ(t) > 0 for some t > 0. Denote by Eφ = Eφ(Bk) the Orlicz space of finite elements (see (1.6)) generated by φ. The aim of this paper is to show that there exists a retraction of the closed unit ball in Eφ onto the unit sphere in Eφ being a (2 + ɛ)γφ;-set contraction (Theorem 3.6), which generalises [9, Corollary 6] proved for the case of Lp[−1, 1], 1 ≤ p < ∞. Here γφ, denote the Hausdorff measure of noncompactness. This theorem is proved both for the Amemiya and the Luxemburg norms. Also some related results concerning the case of s-convex (0 < s ≤ 1) functions are presented.

scholarly journals Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

2019 ◽  
Author(s):  
Florian Besau ◽  
Steven Hoehner ◽  
Gil Kur
Keyword(s):  
Unit Ball ◽  
Best Approximation ◽  
Convex Bodies ◽  
Asymptotic Formulas ◽  
Smooth Convex ◽  
High Dimensions ◽  
Sharp Bounds ◽  
Intrinsic Volumes ◽  
The Best Approximation ◽  
Euclidean Unit Ball

Abstract We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.

scholarly journals TWO NOTABLE CLASSES OF PROJECTIVE VECTOR FIELDS

2020 ◽  
pp. 741
Author(s):  
Tayebeh Tabatabeifar ◽  
Mehdi Rafie-Rad ◽  
Behzad Najafi
Keyword(s):  
Vector Field ◽  
Unit Ball ◽  
Vector Fields ◽  
Necessary Conditions ◽  
Randers Metric ◽  
Projective Vector ◽  
Euclidean Unit Ball

Here, we find some necessary conditions for a projective vector field on a Randers metric to preserve the non-Riemannian quantities $\Xi$ and $H$.They are known in the contexts as the $C$-projective and $H$-projective vector fields. We find all projective vector fields of the Funk type metrics on the Euclidean unit ball $\mathbb{B}^n(1)$. 

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