scholarly journals Hahn-Banach and Sandwich Theorems for Equivariant Vector Lattice-Valued Operators and Applications

2020 ◽  
Vol 76 (1) ◽  
pp. 11-34
Author(s):  
Antonio Boccuto
Keyword(s):  
Vector Lattice ◽  
Fenchel Duality ◽  
Separation Theorems ◽  
Extension Theorems

AbstractWe prove Hahn-Banach, sandwich and extension theorems for vector lattice-valued operators, equivariant with respect to a given group G of homomorphisms. As applications and consequences, we present some Fenchel duality and separation theorems, a version of the Moreau-Rockafellar formula and some Farkas and Kuhn-Tucker-type optimization results. Finally, we prove that the obtained results are equivalent to the amenability of G.

scholarly journals Subdifferential calculus for invariant linear ordered vector space-valued operators and applications

2016 ◽  
Vol 12 (4) ◽  
pp. 6160-6170
Author(s):  
Antonio Boccuto
Keyword(s):  
Vector Space ◽  
Vector Lattice ◽  
Direct Proof ◽  
Extension Theorems ◽  
Subdifferential Calculus ◽  
Partially Ordered ◽  
Ordered Vector Space ◽  
Partially Ordered Vector Space ◽  
Sandwich Type ◽  
Linear Invariant

We give a direct proof of sandwich-type theorems for linear invariant partially ordered vector space operators in the setting of convexity. As consequences, we deduce equivalence results between sandwich, Hahn-Banach, separation and Krein-type extension theorems, Fenchel duality, Farkas and Kuhn-Tucker-type minimization results and subdifferential formulas in the context of invariance. As applications, we give Tarski-type extension theorems and related examples for vector lattice-valued invariant probabilities, defined on suitable kinds of events.

Friedrichs type extension theorems for multivalued operators

1991 ◽  
Vol 2 (1) ◽  
pp. 105-121
Author(s):  
George Dinca ◽  
Daniel Mateescu
Keyword(s):  
Extension Theorems ◽  
Multivalued Operators

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.

General Extension Theorems for Linear Functionals

1974 ◽  
Vol 61 (1) ◽  
pp. 111-122 ◽  
Author(s):  
M. Landsberg ◽  
W. Schirotzek
Keyword(s):  
Linear Functionals ◽  
Extension Theorems

Hahn-Banach-Type Theorems and Subdifferentials for Invariant and Equivariant Order Continuous Vector Lattice-Valued Operators with Applications to Optimization

2021 ◽  
Vol 78 (1) ◽  
pp. 139-156
Author(s):  
Antonio Boccuto
Keyword(s):  
Optimality Conditions ◽  
Vector Lattice ◽  
Optimization Problems ◽  
Linear Constraints ◽  
Continuous Vector ◽  
Fixed Group ◽  
Composite Functions ◽  
Order Continuous

Abstract We give some versions of Hahn-Banach, sandwich, duality, Moreau--Rockafellar-type theorems, optimality conditions and a formula for the subdifferential of composite functions for order continuous vector lattice-valued operators, invariant or equivariant with respect to a fixed group G of homomorphisms. As applications to optimization problems with both convex and linear constraints, we present some Farkas and Kuhn-Tucker-type results.

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