scholarly journals On the average distance property in finite dimensional real Banach spaces

1995 ◽  
Vol 51 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Positive Real Number ◽  
Real Banach Space ◽  
Average Distance ◽  
Positive Real ◽  
Finite Dimensional ◽  
Distance Property ◽  
Average Distance Property ◽  
Image Position

The average distance Theorem of Gross implies that for each N-dimensional real Banach space E (N ≥ 2) there is a unique positive real number r(E) with the following property: for each positive integer n and for all (not necessarily distinct) x1, x2, …, xn, in E with ‖x1‖ = ‖x2‖ = … = ‖xn‖ = 1, there exists an x in E with ‖x‖ = 1 such that.In this paper we prove that if E has a 1-unconditional basis then r(E)≤2−(l/N) and equality holds if and only if E is isometrically isomorphic to Rn equipped with the usual 1-norm.

scholarly journals Average distances in compact connected spaces

1982 ◽  
Vol 26 (3) ◽  
pp. 331-342 ◽  
Author(s):  
David Yost
Keyword(s):  
Convex Sets ◽  
Average Distance ◽  
Compact Convex ◽  
Topological Spaces ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Distance Property ◽  
Compact Convex Set ◽  
Average Distance Property ◽  
Image Position

We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X withWe examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

scholarly journals The average distance property of classical Banach spaces

2000 ◽  
Vol 62 (1) ◽  
pp. 119-134 ◽  
Author(s):  
Aicke Hinrichs
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Real Number ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Average Distance ◽  
Distance Property ◽  
Average Distance Property

A Banach space X has the average distance property (ADP) if there exists a unique real number r such that for each positive integer n and all x1,…,xn in the unit sphere of X there is some x in the unit sphere of X such that .It is known that l2 and l∞ have the ADP, whereas lp fails to have the ADP if 1 ≤ p < 2. We show that lp also fails to have the ADP for 3 ≤ p ≤ ∞. Our method seems to be able to decide also the case 2 < p < 3, but the computational difficulties increase as p comes closer to 2.

scholarly journals The average distance property of classical Banach spaces II

2002 ◽  
Vol 65 (3) ◽  
pp. 511-520 ◽  
Author(s):  
Aicke Hinrichs ◽  
Jörg Wenzel
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Real Number ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Average Distance ◽  
Distance Property ◽  
Average Distance Property

A Banach space X has the average distance property if there exists a unique real number r such that for each positive integer n and all x1,…,xn in the unit sphere of X there is some x in the unit sphere of X such that We show that lp does not have the average distance property if p > 2. This completes the study of the average distance property for lp spaces.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals Problems with Mixed Boundary Conditions in Banach Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Dionicio Pastor Dallos Santos
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Real Number ◽  
Real Banach Space ◽  
Mixed Boundary ◽  
Lipschitz Constant ◽  
Mixed Boundary Conditions ◽  
Positive Real ◽  
Condensing Maps

Using Leray-Schauder degree or degree for α-condensing maps we obtain the existence of at least one solution for the boundary value problem of the following type: φu′′=ft,u,u′,  u(T)=0=u′(0), where φ:X→X is a homeomorphism with reverse Lipschitz constant such that φ(0)=0, f:0,T×X×X→X is a continuous function, T is a positive real number, and X is a real Banach space.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

On the average distance property of compact connected metric spaces

1983 ◽  
Vol 40 (1) ◽  
pp. 459-463 ◽  
Author(s):  
Sidney A. Morris ◽  
Peter Nickolas
Keyword(s):  
Metric Spaces ◽  
Average Distance ◽  
Distance Property ◽  
Average Distance Property

AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

The limit theorem for aperiodic discrete renewal processes

1964 ◽  
Vol 4 (1) ◽  
pp. 122-128
Author(s):  
P. D. Finch
Keyword(s):  
Limit Theorem ◽  
Real Number ◽  
Renewal Process ◽  
Positive Real Number ◽  
Random Variables ◽  
Renewal Processes ◽  
Positive Real ◽  
Image Position

A discrete renewal process is a sequence {X4} of independently and inentically distributed random variables which can take on only those values which are positive integral multiples of a positive real number δ. For notational convenience we take δ = 1 and write where .

Diophantine Approximation and Horocyclic Groups

1957 ◽  
Vol 9 ◽  
pp. 277-290 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Modular Group ◽  
Positive Real Number ◽  
Irrational Number ◽  
Infinitely Many Solutions ◽  
Positive Real ◽  
Closed Set ◽  
Coprime Integers ◽  
Image Position

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality(1)has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

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