scholarly journals The completeness of a normed space is equivalent to the homogeneity of its space of closed bounded convex sets

2013 ◽  
Vol 5 (1) ◽  
pp. 44-46
Author(s):  
I. Hetman
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Infinite Dimensional ◽  
Dimensional Normed Space ◽  
Bounded Convex ◽  
Convex Subsets

We prove that an infinite-dimensional normed space $X$ is complete if and only if the space $\mathrm{BConv}_H(X)$ of all non-empty bounded closed convex subsets of $X$ is topologically homogeneous.

On Covering the Unit Ball in Normed Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 107-109 ◽  
Author(s):  
J. Connett
Keyword(s):  
Unit Ball ◽  
Normed Space ◽  
Normed Spaces ◽  
Finite Covering ◽  
Covering Property ◽  
Infinite Dimensional ◽  
Dimensional Normed Space

By compactness, the unit ball Bn in Rn has a finite covering by translates of rBn, for any r > 0. The main theorem of this note shows that a weaker covering property does not hold in any infinite-dimensional normed space.

scholarly journals Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces

2021 ◽  
Vol 7 (3) ◽  
pp. 3290-3302
Author(s):  
Ruini Li ◽  
◽  
Jianrong Wu
Keyword(s):  
Normed Space ◽  
Convex Cone ◽  
Convex Sets ◽  
Separation Theorems ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Separation Of Convex Sets ◽  
Convex Subsets

<abstract> <p>In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.</p> </abstract>

Balanced and absorbing subsets with empty interior

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Enrique Naranjo-Guerra
Keyword(s):  
Unit Ball ◽  
Normed Space ◽  
Convex Subset ◽  
Linear Span ◽  
Empty Interior ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Real Normed Space ◽  
Whole Space ◽  
Dimensional Normed Space

AbstractOur first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. In our second result we prove that every real or complex separable normed space with dimension greater than 1 contains a balanced and absorbing subset with empty interior which is dense in the unit ball. Explicit constructions of these subsets are given.

Lipschitz mappings on the unit sphere of an infinite-dimensional normed space

2002 ◽  
Vol 79 (5) ◽  
pp. 379-384 ◽  
Author(s):  
A. Jiménez-Vargas ◽  
J. F. Mena-Jurado ◽  
J. C. Navarro-Pascual
Keyword(s):  
Unit Sphere ◽  
Normed Space ◽  
Infinite Dimensional ◽  
Lipschitz Mappings ◽  
Dimensional Normed Space

scholarly journals About an Erdős–Grünbaum Conjecture Concerning Piercing of Non-bounded Convex Sets

2015 ◽  
Vol 53 (4) ◽  
pp. 941-950
Author(s):  
Amanda Montejano ◽  
Luis Montejano ◽  
Edgardo Roldán-Pensado ◽  
Pablo Soberón
Keyword(s):  
Convex Sets ◽  
Bounded Convex

Quotients of Essentially Euclidean Spaces

2019 ◽  
Vol 62 (1) ◽  
pp. 71-74
Author(s):  
Tadeusz Figiel ◽  
William Johnson
Keyword(s):  
Normed Space ◽  
Quotient Space ◽  
Euclidean Spaces ◽  
Quantitative Version ◽  
Finite Dimensional ◽  
Dimensional Normed Space

AbstractA precise quantitative version of the following qualitative statement is proved: If a finite-dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property.

On the Vector Sum of Two Convex Sets in Space

1991 ◽  
Vol 43 (2) ◽  
pp. 347-355 ◽  
Author(s):  
Steven G. Krantz ◽  
Harold R. Parks
Keyword(s):  
Convex Sets ◽  
Analytic Boundary ◽  
Real Analytic ◽  
Vector Sum ◽  
Bounded Convex

In the paper [KIS2], C. Kiselman studied the boundary smoothness of the vector sum of two smoothly bounded convex sets A and B in . He discovered the startling fact that even when A and B have real analytic boundary the set A + B need not have boundary smoothness exceeding C20/3 (this result is sharp). When A and B have C∞ boundaries, then the smoothness of the sum set breaks down at the level C5 (see [KIS2] for the various pathologies that arise).

scholarly journals Valuations on convex functions and convex sets and Monge–Ampère operators

2019 ◽  
Vol 19 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Semyon Alesker
Keyword(s):  
Convex Functions ◽  
Convex Sets ◽  
Convex Bodies ◽  
Linear Functionals ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Dense Image ◽  
Monge Ampere ◽  
Explicit Relation ◽  
Class Of Functions

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

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