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.

scholarly journals The horofunction boundary of finite-dimensional normed spaces

2007 ◽  
Vol 142 (3) ◽  
pp. 497-507 ◽  
Author(s):  
CORMAC WALSH
Keyword(s):  
Unit Ball ◽  
Normed Space ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Dimensional Normed Space

AbstractWe determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of “almost-geodesics”. We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is closed in the Painlevé–Kuratowski topology.

scholarly journals Maximum average distance in complex finite dimensional normed spaces

2002 ◽  
Vol 66 (1) ◽  
pp. 125-134
Author(s):  
Juan C. García-Vázquez ◽  
Rafael Villa
Keyword(s):  
Unit Sphere ◽  
Metric Space ◽  
Normed Space ◽  
Average Distance ◽  
Complex Case ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Real Normed Space ◽  
Dimensional Normed Space

A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xn ∈ M, there exists x ∈ M such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

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 Midpoint sets contained in the unit sphere of a normed space

2011 ◽  
Vol 48 (2) ◽  
pp. 180-192
Author(s):  
Konrad Swanepoel
Keyword(s):  
Unit Ball ◽  
Unit Sphere ◽  
Normed Space ◽  
Higher Dimensions ◽  
Three Dimensions ◽  
Closed Unit Ball ◽  
Facial Structure ◽  
Finite Dimensional ◽  
Dimensional Normed Space

The midpoint set M(S) of a set S of points is the set of all midpoints of pairs of points in S. We study the largest cardinality of a midpoint set M(S) in a finite-dimensional normed space, such that M(S) is contained in the unit sphere, and S is outside the closed unit ball. We show in three dimensions that this maximum (if it exists) is determined by the facial structure of the unit ball. In higher dimensions no such relationship exists. We also determine the maximum for euclidean and sup norm spaces.

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.

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.

Infinite-Dimensional Polyhedrality

2004 ◽  
Vol 56 (3) ◽  
pp. 472-494 ◽  
Author(s):  
Vladimir P. Fonf ◽  
Libor Veselý
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
General Definition ◽  
Infinite Dimensions ◽  
Open Problems ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Isomorphic Classification

AbstractThis paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a polytope if each of its finite-dimensional affine sections is a (standard) polytope.We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).

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