scholarly journals The closed convex hull of certain extreme points

1964 ◽  
Vol 15 (2) ◽  
pp. 256-256 ◽  
Author(s):  
Dwight B. Goodner
Keyword(s):  
Convex Hull ◽  
Extreme Points ◽  
Closed Convex Hull

scholarly journals Some results on isometric composition operators on Lipschitz spaces

2021 ◽  
Author(s):  
Abraham Rueda Zoca
Keyword(s):  
Convex Hull ◽  
Composition Operator ◽  
Metric Spaces ◽  
Composition Operators ◽  
Extreme Points ◽  
Necessary Condition ◽  
Closed Convex Hull ◽  
Lipschitz Spaces ◽  
Complete Characterisation

AbstractGiven two metric spaces M and N we study, motivated by a question of N. Weaver, conditions under which a composition operator $$C_\phi :{\mathrm {Lip}}_0(M)\longrightarrow {\mathrm {Lip}}_0(N)$$ C ϕ : Lip 0 ( M ) ⟶ Lip 0 ( N ) is an isometry depending on the properties of $$\phi $$ ϕ . We obtain a complete characterisation of those operators $$C_\phi $$ C ϕ in terms of a property of the function $$\phi $$ ϕ in the case that $$B_{{\mathcal {F}}(M)}$$ B F ( M ) is the closed convex hull of its preserved extreme points. Also, we obtain necessary condition for $$C_\phi $$ C ϕ being an isometry in the case that M is geodesic.

scholarly journals Coefficients of an analytic function subordination class determined by rotations

1996 ◽  
Vol 60 (2) ◽  
pp. 245-254
Author(s):  
Seok Chan Kim
Keyword(s):  
Analytic Function ◽  
Convex Hull ◽  
Extreme Points ◽  
Closed Convex Hull ◽  
Image Position ◽  
Positive Integers ◽  
Compact Subsets

AbstractLet A denote the set of all functions analytic in U = {z: |;z| < 1} equipped with the topology of unifrom convergence on compact subsets of U. For F ∈ A define Let s(F) and s(F) denote the closed convex hull of s(F) and the set of extreme points of , respectively. Let R denote the class of all F ∈ A such that = {Fx}: |x| = 1} where Fx = F(xz).We prove that |An| ≤ |AMN| for all positive integers M and N, and for . We also prove that if , then F is a univelaent halfplane mapping.

Some more characterizations of Banach spaces containing l1

1976 ◽  
Vol 80 (2) ◽  
pp. 269-276 ◽  
Author(s):  
Richard Haydon
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Convex Subset ◽  
Compact Convex ◽  
Extreme Points ◽  
Compact Convex Subset ◽  
Closed Convex Hull ◽  
Separability Assumption

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.

On the Krein-Milman theorem in CAT(κ) spaces

2018 ◽  
Vol 34 (3) ◽  
pp. 401-404
Author(s):  
BANCHA PANYANAK ◽  
Keyword(s):  
Convex Hull ◽  
Convex Subset ◽  
Compact Convex ◽  
Extreme Points ◽  
General Setting ◽  
Compact Convex Subset ◽  
Closed Convex Hull ◽  
Nonempty Compact ◽  
Nonempty Compact Convex Subset

Let κ > 0 and (X, ρ) be a complete CAT(κ) space whose diameter smaller than ... It is shown that if K is a nonempty compact convex subset of X, then K is the closed convex hull of its set of extreme points. This is an extension of the Krein-Milman theorem to the general setting of CAT(κ) spaces.

The Min-Max Load Criteria as a Measure of Machining Fixture Performance

1994 ◽  
Vol 116 (4) ◽  
pp. 500-507 ◽  
Author(s):  
E. C. DeMeter
Keyword(s):  
Convex Hull ◽  
Linear Model ◽  
Model Validation ◽  
Contact Region ◽  
Extreme Points ◽  
Simulation Experiments ◽  
Machining Operations ◽  
The Impact ◽  
External Loads ◽  
The Continuum

Spherical-tipped locators and clamps are often used for the restraint of castings during machining. For structurally rigid castings, contact region deformation and micro-slippage are the predominant modes of workpiece displacement. In turn contact region deformation and micro-slippage are heavily influenced by contact region loading. This paper presents a linear model for predicting the impact of locator and clamp placement on workpiece displacement throughout a series of machining operations. It illustrates how the continuum of external loads exerted on a workpiece during machining can be bounded within a convex hull, and how the extreme points of this hull are used within the model. Finally it describes the simulation experiments which were used for model validation.

Digital Modeling and Movement Resolution of Rubber Band Analogy for 2D Packing Problem

2012 ◽  
Vol 220-223 ◽  
pp. 2466-2470 ◽  
Author(s):  
Jun Yan Ma ◽  
Xiao Ping Liao ◽  
Juan Lu ◽  
Hong Yao
Keyword(s):  
Convex Hull ◽  
Extreme Points ◽  
Packing Problem ◽  
Rubber Band ◽  
Scanning Method ◽  
Resultant Vector ◽  
Digital Modeling ◽  
Volume Decrease ◽  
Modeling Algorithm

Packing problem is how to arrange the components in available spaces to make the layout compact. This paper adopts a digital modeling algorithm to establish a novel rubber band convex hull model to solve this problem. A ray scanning method analogy QuickHull algorithm is presented to get extreme points of rubber band convex hull. A plural vector expression approach is adopted to movement resolution,which calculate the resultant vector to translate, rotate and slide the subbody to make the volume decrease. An experiment proved this digital modeling algorithm effective.

Ε‎-Fr ´Echet Differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Simple Proof ◽  
Common Point ◽  
Lipschitz Functions ◽  
Closed Convex Hull ◽  
Uniformly Smooth ◽  
Fréchet Differentiability ◽  
Finite Set ◽  
Frechet Differentiability

This chapter treats results on ε‎-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives. The chapter first presents a simple proof of an almost differentiability result for Lipschitz functions in asymptotically uniformly smooth spaces before discussing the notion of asymptotic uniform smoothness. It then proves that in an asymptotically smooth Banach space X, any finite set of real-valued Lipschitz functions on X has, for every ε‎ > 0, a common point of ε‎-Fréchet differentiability.

The number of extreme points in the convex hull of a random sample

1991 ◽  
Vol 28 (02) ◽  
pp. 287-304 ◽  
Author(s):  
David J. Aldous ◽  
Bert Fristedt ◽  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Convex Hull ◽  
Random Sample ◽  
Limit Distribution ◽  
Extreme Points ◽  
Radial Component ◽  
Limiting Distribution ◽  
Precise Information ◽  
Mean And Variance ◽  
The Mean

Let {Xk} be an i.i.d. sequence taking values in ℝ2with the radial and spherical components independent and the radial component having a distribution with slowly varying tail. The number of extreme points in the convex hull of {X1, · ··,Xn} is shown to have a limiting distribution which is obtained explicitly. Precise information about the mean and variance of the limit distribution is obtained.

scholarly journals EXTREME POINTS OF INTEGRAL FAMILIES OF ANALYTIC FUNCTIONS

2013 ◽  
Vol 94 (2) ◽  
pp. 202-221
Author(s):  
KEIKO DOW ◽  
D. R. WILKEN
Keyword(s):  
Analytic Functions ◽  
Compact Convex ◽  
Extreme Points ◽  
Starlike Functions ◽  
Extremal Problems ◽  
Kernel Functions ◽  
Closed Convex Hull ◽  
New Class ◽  
Geometric Function ◽  
Derivatives Of

AbstractExtreme 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 two-parameter collection of kernel functions integrated against measures on the torus. For specific choices of the parameters many families from classical geometric function theory are included. These families include 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. The main result introduces a surprising new class of extreme points.

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