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 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.

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.

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 Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

scholarly journals Remarks on retracting balls on spherical caps in \(c_{0}\), \(c\), \(l^{\infty }\) spaces

2020 ◽  
Vol 74 (1) ◽  
pp. 45
Author(s):  
Kazimierz Goebel
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Unit Sphere ◽  
Uniform Norm ◽  
Sequence Spaces ◽  
Closed Unit Ball ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Lipschitz Constants ◽  
Spherical Caps

For any infinite dimensional Banach space there exists a lipschitzian retraction of the closed unit ball B onto the unit sphere S. Lipschitz constants for such retractions are, in general, only roughly estimated. The paper is illustrative. It contains remarks, illustrations and estimates concerning optimal retractions onto spherical caps for sequence spaces with the uniform norm.

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.

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