The unit balls of ℒ(nl∞m) and ℒs(nl∞m)

2020 ◽  
Vol 57 (3) ◽  
pp. 267-283
Author(s):  
Sung Guen Kim
Keyword(s):  
Extreme Point ◽  
Extreme Points ◽  
Supremum Norm ◽  
Linear Forms

AbstractFor n,m≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of and , where is the space of n-linear forms on with the supremum norm, and is the subspace of consisting of symmetric n-linear forms. First we classify the extreme points of the unit balls of and , respectively. We show that ext ⊂ ext , which answers the question in [32]. We show that every extreme point of the unit balls of and is exposed, correspondingly. We also show thatand which answers the questions in [31].

Some Extreme Rays of the Positive Pluriharmonic Functions

1979 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Ball ◽  
Extreme Point ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Compact Open Topology ◽  
Open Unit Ball ◽  
Pluriharmonic Functions ◽  
Image Position ◽  
Extreme Rays

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. LetThus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if(1.1)if(1.2)and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if(1.3)

Orlicz Spaces Without Strongly Extreme Points and Without H-Points

1993 ◽  
Vol 36 (2) ◽  
pp. 173-177 ◽  
Author(s):  
Henryk Hudzik
Keyword(s):  
Extreme Point ◽  
Orlicz Space ◽  
Unit Sphere ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Luxemburg Norm ◽  
Nonatomic Measure

AbstractW. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

scholarly journals Geometry of multilinear forms

2019 ◽  
Vol 22 (02) ◽  
pp. 1950011 ◽  
Author(s):  
W. V. Cavalcante ◽  
D. M. Pellegrino ◽  
E. V. Teixeira
Keyword(s):  
Unit Ball ◽  
Extreme Points ◽  
Linear Forms ◽  
Multilinear Forms ◽  
Elementary Steps ◽  
Full Characterization ◽  
Constructive Process

We develop a constructive process which determines all extreme points of the unit ball in the space of [Formula: see text]-linear forms, [Formula: see text] Our method provides a full characterization of the geometry of that space through finitely many elementary steps, and thus it can be extensively applied in both computational as well as theoretical problems; few consequences are also derived in this paper.

scholarly journals Strongly Extreme Points and Approximation Properties

2018 ◽  
Vol 61 (3) ◽  
pp. 449-457
Author(s):  
Trond A. Abrahamsen ◽  
Petr Hájek ◽  
Olav Nygaard ◽  
Stanimir L. Troyanski
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Convex Subset ◽  
Approximation Property ◽  
Extreme Points ◽  
Local Approximation ◽  
Approximation Properties ◽  
Compact Approximation ◽  
Approximation Of The Identity

AbstractWe show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

scholarly journals Non-extreme-Points Approach to Extreme Points of Integral Families of Analytic Functions

10.53733/87 ◽  
2021 ◽  
Vol 51 ◽  
pp. 39-48
Author(s):  
Keiko Dow
Keyword(s):  
Unit Circle ◽  
Extreme Point ◽  
Analytic Functions ◽  
Compact Convex ◽  
Extreme Points ◽  
Starlike Functions ◽  
Extremal Problems ◽  
Kernel Functions ◽  
Closed Convex Hull ◽  
Special Cases

Non extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a 2-parameter collection of kernel functions integrated against measures on the torus. Families from classical geometric function theory such as the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others are included. However for these families of analytic functions, identifying “all” the extreme points remains a difficult challenge except in some special cases. Aharonov and Friedland [1] identified a band of points on the unit circle which corresponds to the set of extreme points for these 2-parameter collections of kernel functions. Later this band of extreme points was further extended by introducing a new technique by Dow and Wilken [3]. On the other hand, a technique to identify a non extreme point was not investigated much in the past probably because identifying non extreme points does not directly help solving the optimization of linear extremal problems. So far only one point on the unit circle has beenidentified which corresponds to a non extreme point for a 2-parameter collections of kernel functions. This leaves a big gap between the band of extreme points and one non extreme point. The author believes it is worth developing some techniques, and identifying non extreme points will shed a new light in the exact determination of the extreme points. The ultimate goal is to identify the point on the unit circle that separates the band of extreme points from non extreme points. The main result introduces a new class of non extreme points.

scholarly journals The norming set of a symmetric bilinear form on the plane with the supremum norm

2021 ◽  
Vol 55 (2) ◽  
pp. 171-180
Author(s):  
S. G. Kim
Keyword(s):  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Supremum Norm ◽  
Linear Forms

An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}_s(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and$|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}_s(^n E)$ denotes the space of all symmetric continuous $n$-linear forms on $E.$For $T\in {\mathcal L}_s(^n E),$ we define $$\mathop{\rm Norm}(T)=\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\}.$$$\mathop{\rm Norm}(T)$ is called the {\em norming set} of $T$. We classify $\mathop{\rm Norm}(T)$ for every $T\in {\mathcal L}_s(^2l_{\infty}^2)$.

scholarly journals Extreme Point Methods and Banach-Stone Theorems

2003 ◽  
Vol 75 (1) ◽  
pp. 125-143 ◽  
Author(s):  
Hasan Al-Halees ◽  
Richard J. Fleming
Keyword(s):  
Extreme Point ◽  
Hausdorff Space ◽  
Weighted Composition Operator ◽  
Extreme Points ◽  
Continuous Functions ◽  
Spaces Of Continuous Functions ◽  
Weighted Composition ◽  
Function Module ◽  
Nice Operator ◽  
Vector Valued

AbstractAn operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of C0( Q, X) into C0(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.

scholarly journals Extreme Points and Rotundity in Musielak-Orlicz-Bochner Function Spaces Endowed with Orlicz Norm

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoqiang Shang ◽  
Yunan Cui ◽  
Yongqiang Fu
Keyword(s):  
Extreme Point ◽  
Orlicz Function ◽  
Extreme Points ◽  
Function Spaces ◽  
Orlicz Norm

The criteria for extreme point and rotundity of Musielak-Orlicz-Bochner function spaces equipped with Orlicz norm are given. Although criteria for extreme point of Musielak-Orlicz function spaces equipped with the Orlicz norm were known, we can easily deduce them from our main results.

scholarly journals Extreme points of ${\mathcal L}_s(^2l_{\infty})$ and ${\mathcal P}(^2l_{\infty})$

2021 ◽  
Vol 13 (2) ◽  
pp. 289-297
Author(s):  
Sung Guen Kim
Keyword(s):  
Unit Ball ◽  
Extreme Point ◽  
Extreme Points

For $n\geq 2,$ we show that every extreme point of the unit ball of ${\mathcal L}_s(^2l_{\infty}^n)$ is extreme in ${\mathcal L}_s(^2l_{\infty}^{n+1})$, which answers the question in [Period. Math. Hungar. 2018, 77 (2), 274-290]. As a corollary we show that every extreme point of the unit ball of ${\mathcal L}_s(^2l_{\infty}^n)$ is extreme in ${\mathcal L}_s(^2l_{\infty})$. We also show that every extreme point of the unit ball of ${\mathcal P}(^2l_{\infty}^2)$ is extreme in ${\mathcal P}(^2l_{\infty}^n).$ As a corollary we show that every extreme point of the unit ball of ${\mathcal P}(^2l_{\infty}^2)$ is extreme in ${\mathcal P}(^2l_{\infty})$.

(D-2)-Extreme Points and a Helly-Type Theorem for Starshaped Sets

1980 ◽  
Vol 32 (3) ◽  
pp. 703-713 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Extreme Point ◽  
Type Theorem ◽  
Extreme Points ◽  
Compact Set ◽  
Dimensional Simplex ◽  
Starshaped Set ◽  
Starshaped Sets

We begin with some preliminary definitions. Let S be a subset of Rd. For points x and y in S, we say x sees y via S if and only if the corresponding segment [x, y] lies in S. The set Sis said to be starshaped if and only if there is some point p in S such that, for every x in S, p sees x via S. The collection of all such points p is called the kernel of S, denoted ker S. Furthermore, if we define the star of x in S by Sx = {y: [x, y] ⊆ S}, it is clear that ker S = ⋂{Sx: x in S}.Several interesting results indicate a relationship between ker S and the set E of (d – 2)-extreme points of S. Recall that for d ≧ 2, a point x in S is a (d – 2)-extreme point of S if and only if x is not relatively interior to a (d – 1)-dimensional simplex which lies in S. Kenelly, Hare et al. [4] have proved that if S is a compact starshaped set in Rd, d ≧ 2, then ker S = ⋂{Se: eE}. This was strengthened in papers by Stavrakas [6] and Goodey [2], and their results show that the conclusion follows whenever S is a compact set whose complement ~S is connected.

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