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 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.

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 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.

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.

scholarly journals HYPERPLANES OF FINITE-DIMENSIONAL NORMED SPACES WITH THE MAXIMAL RELATIVE PROJECTION CONSTANT

2015 ◽  
Vol 91 (3) ◽  
pp. 447-463 ◽  
Author(s):  
TOMASZ KOBOS
Keyword(s):  
Normed Space ◽  
Upper Bound ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Subspace ◽  
Equivalent Condition ◽  
Normed Spaces ◽  
Codimension One ◽  
Projection Constant ◽  
Dimensional Normed Space

The relative projection constant${\it\lambda}(Y,X)$ of normed spaces $Y\subset X$ is ${\it\lambda}(Y,X)=\inf \{\Vert P\Vert :P\in {\mathcal{P}}(X,Y)\}$, where ${\mathcal{P}}(X,Y)$ denotes the set of all continuous projections from $X$ onto $Y$. By the well-known result of Bohnenblust, for every $n$-dimensional normed space $X$ and a subspace $Y\subset X$ of codimension one, ${\it\lambda}(Y,X)\leq 2-2/n$. The main goal of the paper is to study the equality case in the theorem of Bohnenblust. We establish an equivalent condition for the equality ${\it\lambda}(Y,X)=2-2/n$ and present several applications. We prove that every three-dimensional space has a subspace with the projection constant less than $\frac{4}{3}-0.0007$. This gives a nontrivial upper bound in the problem posed by Bosznay and Garay. In the general case, we give an upper bound for the number of ($n-1$)-dimensional subspaces with the maximal relative projection constant in terms of the facets of the unit ball of $X$. As a consequence, every $n$-dimensional normed space $X$ has an ($n-1$)-dimensional subspace $Y$ with ${\it\lambda}(Y,X)<2-2/n$. This contrasts with the separable case in which it is possible that every hyperplane has a maximal possible projection constant.

Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls

2014 ◽  
Vol 57 (1) ◽  
pp. 42-50 ◽  
Author(s):  
Vladimir P. Fonf ◽  
Clemente Zanco
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Finite Covering ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractWe prove that, given any covering of any infinite-dimensional Hilbert space H by countably many closed balls, some point exists in H which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.

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.

scholarly journals Uniformly invariant normed spaces

BIBECHANA ◽  
2013 ◽  
Vol 10 ◽  
pp. 31-33
Author(s):  
AM Forouzanfar ◽  
S Khorshidvandpour ◽  
Z Bahmani
Keyword(s):  
Hilbert Space ◽  
Positive Operator ◽  
Normed Space ◽  
Open Problem ◽  
Closed Subspace ◽  
Normed Spaces ◽  
Dimensional Normed Space

In this work, we introduce the concepts of compactly invariant and uniformly invariant. Also we define sometimes C-invariant closed subspaces and then prove every m-dimensional normed space with m > 1 has a nontrivial sometimes C-invariant closed subspace. Sequentially C-invariant closed subspaces are also introduced. Next, An open problem on the connection between compactly invariant and uniformly invariant normed spaces has been posed. Finally, we prove a theorem on the existence of a positive operator on a strict uniformly invariant Hilbert space. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.7555 BIBECHANA 10 (2014) 31-33

On Nearly Equilateral Simplices and Nearly l∞ Spaces

2010 ◽  
Vol 53 (3) ◽  
pp. 394-397 ◽  
Author(s):  
Gennadiy Averkov
Keyword(s):  
Metric Space ◽  
Normed Space ◽  
Normed Spaces ◽  
Dimensional Normed Space

AbstractBy d(X, Y) we denote the (multiplicative) Banach–Mazur distance between two normed spaces X and Y. Let X be an n-dimensional normed space with d(X, ) ≤ 2, where stands for ℝn endowed with the norm ║(x1, … , xn)║∞ := max﹛|x1|, … , |xn|﹜. Then every metric space (S, ρ) of cardinality n + 1 with norm ρ satisfying the condition maxD/minD ≤ 2/ d(X, ) for D := ﹛ρ(a, b) : a, b ∈ S, a ≠ b﹜ can be isometrically embedded into X.

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