A characterization of convex sets via visibility

2002 ◽  
Vol 64 (1-2) ◽  
pp. 128-135 ◽  
Author(s):  
H. Martini ◽  
W. Wenzel
Keyword(s):  
Convex Sets

scholarly journals Theoretical framework for higher-order quantum theory

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180706 ◽  
Author(s):  
Alessandro Bisio ◽  
Paolo Perinotti
Keyword(s):  
Quantum Theory ◽  
Convex Sets ◽  
Mathematical Structure ◽  
Higher Order ◽  
Equivalence Relations ◽  
Full Generality ◽  
Special Cases ◽  
Different Types ◽  
Causal Structures

Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive characterization of convex sets of maps of a given type is used to prove equivalence relations between different types. The axioms of the framework do not refer to the specific mathematical structure of quantum theory, and can therefore be exported in the context of any operational probabilistic theory.

A Characterization of Finite Dimensional Convex Sets

1952 ◽  
Vol 74 (3) ◽  
pp. 683 ◽  
Author(s):  
E. G. Straus ◽  
F. A. Valentine
Keyword(s):  
Convex Sets ◽  
Finite Dimensional

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.)

scholarly journals A Characterization of Polynomially Convex Sets in Banach Spaces

2017 ◽  
Vol 72 (4) ◽  
pp. 2013-2021
Author(s):  
Mortaza Abtahi ◽  
Sara Farhangi
Keyword(s):  
Banach Spaces ◽  
Convex Sets ◽  
Polynomially Convex

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.

scholarly journals ON SOME GENERALIZATION OF SMOOTHING PROBLEMS

2015 ◽  
Vol 20 (3) ◽  
pp. 311-328 ◽  
Author(s):  
Svetlana Asmuss ◽  
Natalja Budkina
Keyword(s):  
Hilbert Spaces ◽  
Convex Sets ◽  
Special Cases

The paper deals with the generalized smoothing problem in abstract Hilbert spaces. This generalized problem involves particular cases such as the interpolating problem, the smoothing problem with weights, the smoothing problem with obstacles, the problem on splines in convex sets and others. The theorem on the existence and characterization of a solution of the generalized problem is proved. It is shown how the theorem gives already known theorems in special cases as well as some new results.

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.

Sign in / Sign up

Export Citation Format

Share Document