TOPOLOGICAL TOMOGRAPHY IN CONVEXITY

2002 ◽  
Vol 34 (3) ◽  
pp. 353-358
Author(s):  
L. MONTEJANO ◽  
E. SHCHEPIN
Keyword(s):  
Linear Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Weakly Closed ◽  
Locally Convex

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersections, to arbitrary weakly closed subsets of a locally convex linear space.

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

A geometric characterization concerning compact, convex sets

1991 ◽  
Vol 109 (2) ◽  
pp. 351-361 ◽  
Author(s):  
Christopher J. Mulvey ◽  
Joan Wick Pelletier
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Necessary And Sufficient Condition ◽  
Categorical Logic ◽  
Compact Convex Set ◽  
Necessary And Sufficient ◽  
Formal Statement

In this paper, we are concerned with establishing a characterization of any compact, convex set K in a normed space A in an arbitrary topos with natural number object. The characterization is geometric, not in the sense of categorical logic, but in the intuitive one, of describing any compact, convex set K in terms of simpler sets in the normed space A. It is a characterization of the compact, convex set in the sense that it provides a necessary and sufficient condition for an element of the normed space to lie within it. Having said this, we should immediately qualify our statement by stressing that this is the intuitive content of what is proved; the formal statement of the characterization is required to be in terms appropriate to the constructive context of the techniques used.

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

2013 ◽  
Vol 56 (2) ◽  
pp. 272-282 ◽  
Author(s):  
Lixin Cheng ◽  
Zhenghua Luo ◽  
Yu Zhou
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Lower Semicontinuous ◽  
Weakly Compact ◽  
Bounded Set ◽  
Fréchet Differentiable ◽  
Bounded Convex

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.

scholarly journals Quotients of F-spaces

1978 ◽  
Vol 19 (2) ◽  
pp. 103-108 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Analytic Function ◽  
Linear Space ◽  
Quotient Space ◽  
Closed Subspace ◽  
Weakly Closed ◽  
Locally Convex

Let X be a non-locally convex F-space (complete metric linear space) whose dual X′ separates the points of X. Then it is known that X possesses a closed subspace N which fails to be weakly closed (see [3]), or, equivalently, such that the quotient space X/N does not have a point separating dual. However the question has also been raised by Duren, Romberg and Shields [2] of whether X possesses a proper closed weakly dense (PCWD) subspace N, or, equivalently a closed subspace N such that X/N has trivial dual. In [2], the space Hp (0<p<1) was shown to have a PCWD subspace; later in [9], Shapiro showed that ℓp (0<p<1) and certain spaces of analytic function have PCWD subspaces. Our first result in this note is that every separable non-locally convex F-space with separating dual has a PCWD subspace.

Inner structure in real vector spaces

2020 ◽  
Vol 27 (3) ◽  
pp. 361-366 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Enrique Naranjo-Guerra
Keyword(s):  
Convex Sets ◽  
Vector Spaces ◽  
Internal Point ◽  
Real Vector ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Intrinsic Characterization ◽  
The One ◽  
Locally Convex

AbstractInternal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifolds, which was not possible by using internal points. We also characterize infinite dimensional real vector spaces by means of the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincides with the one of internal points.

The Embedding of Compact Convex Sets in Locally Convex Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 449-454 ◽  
Author(s):  
James W. Roberts
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Locally Convex Spaces ◽  
Hausdorff Topological Vector Space ◽  
Compact Convex Set ◽  
Affine Functions ◽  
Locally Convex

In studying compact convex sets it is usually assumed that the compact convex set X is contained in a Hausdorff topological vector space L where the topology on X is the relative topology. Usually one assumes that L is locally convex. The reason for this is that most of the major theorems such as the Krein-Milman, Choquet-Bishop-de Leeuw, and most of the fixed point theorems require that there be enough continuous affine functions on X to separate points.

scholarly journals The range of an o-weakly compact mapping

1985 ◽  
Vol 39 (3) ◽  
pp. 391-399 ◽  
Author(s):  
P. G. Dodds
Keyword(s):  
Convex Set ◽  
Vector Measure ◽  
Compact Convex ◽  
Locally Convex Space ◽  
Weakly Compact ◽  
Compact Mapping ◽  
Compact Convex Set ◽  
Locally Convex ◽  
Order Continuous

AbstractIt is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

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