Existence and characterization of minima of concave functions on unbounded convex sets

1985 ◽  
pp. 88-92 ◽  
Author(s):  
Arthur F. Veinott
Keyword(s):  
Convex Sets ◽  
Concave Functions ◽  
Unbounded Convex

PROJECTIONS OF LOG-CONCAVE FUNCTIONS

2012 ◽  
Vol 14 (05) ◽  
pp. 1250036
Author(s):  
ALEXANDER SEGAL ◽  
BOAZ A. SLOMKA
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Duality Mapping ◽  
Concave Functions ◽  
Finite Dimensional ◽  
Lower Semi ◽  
Natural Notion ◽  
Linear Sections ◽  
Funct Anal

Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 3366–3389] that the well-known duality mapping on the class of closed convex sets in ℝn containing the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were recently uniquely extended to this class by Artstein-Avidan and Milman, a natural notion of projection arises. This notion of projection is justified by our result. As a consequence of our main result, we prove that, on the class of lower semi continuous non-negative convex functions attaining 0 at the origin, the polarity operation is the only operation interchanging addition with geometric inf-convolution and the support function is the only operation interchanging addition with inf-convolution.

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.

scholarly journals An embedding theorem for unbounded convex sets in a Banach space

2009 ◽  
Vol 42 (4) ◽  
Author(s):  
Jakub Bielawski ◽  
Jacek Tabor
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Embedding Theorem ◽  
Unbounded Convex

AbstractLet

Congruence normality: The characterization of the doubling class of convex sets

1994 ◽  
Vol 31 (3) ◽  
pp. 397-406 ◽  
Author(s):  
Alan Day
Keyword(s):  
Convex Sets

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

scholarly journals A ``hidden'' characterization of approximatively polyhedral convex sets in Banach spaces

2012 ◽  
Vol 210 (2) ◽  
pp. 137-157
Author(s):  
Taras Banakh ◽  
Ivan Hetman
Keyword(s):  
Banach Spaces ◽  
Convex Sets ◽  
Polyhedral Convex ◽  
Polyhedral Convex Sets
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