scholarly journals On denseness of certain norms in Banach spaces

1996 ◽  
Vol 54 (2) ◽  
pp. 183-196 ◽  
Author(s):  
M. Jiménez Sevilla ◽  
J.P. Moreno
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Banach Spaces ◽  
Convex Bodies ◽  
Intersection Property ◽  
Closed Convex Hull ◽  
Locally Linear

We give several results dealing with denseness of certain classes of norms with many vertex points. We prove that, in Banach spaces with the Mazur or the weak* Mazur intersection property, every ball (convex body) can be uniformly approximated by balls (convex bodies) being the closed convex hull of their strongly vertex points. We also prove that given a countable set F, every norm can be uniformly approximated by norms which are locally linear at each point of F.

scholarly journals Delta- and Daugavet points in Banach spaces

2020 ◽  
Vol 63 (2) ◽  
pp. 475-496
Author(s):  
T. A. Abrahamsen ◽  
R. Haller ◽  
V. Lima ◽  
K. Pirk
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Closed Convex Hull ◽  
Müntz Spaces ◽  
Daugavet Property ◽  
Diameter Two Property

AbstractA Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same in L1-spaces, in L1-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the closed convex hull of its Δ-points. This gives rise to a new diameter-two property that we call the convex diametral diameter-two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.

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.

scholarly journals Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010 ◽  
Vol 42 (3) ◽  
pp. 605-619 ◽  
Author(s):  
I. Bárány ◽  
F. Fodor ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Lower And Upper Bounds ◽  
Covering Theorem ◽  
Intrinsic Volumes ◽  
Large Numbers ◽  
Distribution Matching

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

scholarly journals Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010 ◽  
Vol 42 (03) ◽  
pp. 605-619
Author(s):  
I. Bárány ◽  
F. Fodor ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Lower And Upper Bounds ◽  
Covering Theorem ◽  
Intrinsic Volumes ◽  
Large Numbers ◽  
Distribution Matching

LetKbe ad-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote byKnthe convex hull ofnpoints chosen randomly and independently fromKaccording to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of thesth intrinsic volumesVs(Kn) ofKnfors∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes ofKn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

A ξ-weak Grothendieck compactness principle

2021 ◽  
pp. 1-16
Author(s):  
KEVIN BEANLAND ◽  
RYAN M. CAUSEY
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Compact Set ◽  
Closed Convex Hull ◽  
Compact Sets ◽  
Weakly Compact ◽  
Schur Property ◽  
Compactness Principle ◽  
Weakly Compact Sets

Abstract For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

scholarly journals Inequalities for the Surface Area of Projections of Convex Bodies

2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Maximal Surface ◽  
Projection Body ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

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.

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.

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