A Banach Space Whose Elements are Classes of Sets of Constant Width

1975 ◽  
Vol 18 (5) ◽  
pp. 679-689 ◽  
Author(s):  
J. E. Lewis
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Compact Subset ◽  
Compact Convex ◽  
Constant Width ◽  
The Real ◽  
Real Euclidean Space ◽  
Convex Subsets

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

scholarly journals MULTIDIMENSIONAL HYBRID BOUNDARY VALUE PROBLEM

2018 ◽  
Vol 58 (6) ◽  
pp. 402-413
Author(s):  
Marzena Szajewska ◽  
Agnieszka Maria Tereszkiewicz
Keyword(s):  
Euclidean Space ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Special Functions ◽  
Boundary Value ◽  
Fundamental Region ◽  
The Real ◽  
Real Euclidean Space

The purpose of this paper is to discuss three types of boundary conditions for few families of special functions orthogonal on the fundamental region. Boundary value problems are considered on a simplex F in the real Euclidean space Rn of dimension n > 2.

Some extremal properties of convex sets

1975 ◽  
Vol 77 (3) ◽  
pp. 515-524 ◽  
Author(s):  
J. N. Lillington
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Convex Sets ◽  
Compact Convex ◽  
Upper And Lower Bounds ◽  
Arbitrary Convex ◽  
Extremal Properties ◽  
Convex Subsets

In this paper we shall suppose that all convex sets are compact convex subsets of Euclidean space En. We shall be concerned in producing upper and lower bounds for the ‘total edge lengths’ of simplices which are contained in or contain arbitrary convex sets in terms of the inradii and circumradii of these sets. However, before proceeding further, we shall introduce some notation and give some motivation for this work.

Quantizations of Flag Manifolds and Conformal Space Time

1997 ◽  
Vol 09 (04) ◽  
pp. 453-465 ◽  
Author(s):  
R. Fioresi
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Conformal Group ◽  
Flag Manifolds ◽  
The Real ◽  
Rham Complex ◽  
De Rham ◽  
Big Cell ◽  
Real Complex ◽  
Real Euclidean Space

In this paper we work out the deformations of some flag manifolds and of complex Minkowski space viewed as an affine big cell inside G(2,4). All the deformations come in tandem with a coaction of the appropriate quantum group. In the case of the Minkowski space this allows us to define the quantum conformal group. We also give two involutions on the quantum complex Minkowski space, that respectively define the real Minkowski space and the real euclidean space. We also compute the quantum De Rham complex for both real (complex) Minkowski and euclidean space.

scholarly journals The non-emptiness of joint spectral subsets of euclidean n-space

1989 ◽  
Vol 47 (2) ◽  
pp. 300-306 ◽  
Author(s):  
W. J. Ricker ◽  
A. R. Schep
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Joint Spectrum ◽  
Spectral Set ◽  
Continuous Linear Operators ◽  
Real Euclidean Space

AbstractA.McIntosh and A. Pryde introduced and gave some applications of notion of “spectral set”, γ(T), associated with each finite, commuting family of continuous linear operators T in a Banach space. Unlike most concepts of joint spectrum, the set γ(T) is part of real Euclidean space. It is shown that γ(T) is always non-empty whenver there are at least two operators in T.

Convergence of multivalued mils and pramarts in spaces without the RNP

2003 ◽  
Vol 40 (1-2) ◽  
pp. 13-31
Author(s):  
G. Krupa
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Compact Convex ◽  
Property A ◽  
Convergence Results ◽  
Nikodym Property ◽  
Convex Subsets

New convergence results for multivalued martingales in the limit and super- and subpramarts whose values are weakly or strongly compact convex subsets of a separable Banach space are presented. In contrast to the previous results the underlying Banach space does not have the Radon-Nikodym property. A connection between multivalued and single-valued subpramarts is established. New onvergence results for single-valued mils and super- and subpramarts follow from the multivalued results presented here.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

scholarly journals Hyperspaces of max-plus convex subsets of powers of the real line

2012 ◽  
Vol 394 (2) ◽  
pp. 481-487 ◽  
Author(s):  
Lidia Bazylevych ◽  
Dušan Repovš ◽  
Mykhailo Zarichnyi
Keyword(s):  
Real Line ◽  
The Real ◽  
Convex Subsets

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

scholarly journals Decompositions of Weakly Compact Valued Integrable Multifunctions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 863 ◽  
Author(s):  
Luisa Di Piazza ◽  
Kazimierz Musiał
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Compact Convex ◽  
Weakly Compact ◽  
Seminal Paper ◽  
Decomposition Property ◽  
Short Overview ◽  
Decomposition Theorems ◽  
State Of Art

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic.

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