A New Proof of the Lyapunov Convexity Theorem

1990 ◽  
Vol 28 (2) ◽  
pp. 478-481 ◽  
Author(s):  
Fabio Tardella
Keyword(s):  
Convexity Theorem ◽  
Lyapunov Convexity Theorem

scholarly journals Optimization problems for set functions

2001 ◽  
Vol 26 (6) ◽  
pp. 371-383
Author(s):  
Slawomir Dorosiewicz
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Formal Definition ◽  
Convexity Theorem ◽  
First Order ◽  
Set Functions ◽  
Measurable Functions ◽  
Lyapunov Convexity Theorem ◽  
Definition Of ◽  
First Order Optimality Conditions

This paper gives the formal definition of a class of optimization problems, that is, problems of finding conditional extrema of given set-measurable functions. It also formulates the generalization of Lyapunov convexity theorem which is used in the proof of first-order optimality conditions for the mentioned class of optimization problems.

A New Proof of an Inequality of Heinz

1964 ◽  
Vol 7 (1) ◽  
pp. 97-100
Author(s):  
P. S. Bullen
Keyword(s):  
Convexity Theorem ◽  
Heinz Inequality ◽  
Measure Spaces

In a recent paper, [l], Dixmier has proved Heinz' inequality by deducing it from a lemma due to Thorin. In this note it is proved directly from a convexity theorem.Let(M(k), ℳ(k), μ(k)), k = 0, …, n, be measure spaces and Lq(k) (M(k), ℳ(k), μ(k)) be all the functions on M(k) such that

scholarly journals Properties of the trajectories of set-valued integrals in banach spaces

1990 ◽  
Vol 41 (2) ◽  
pp. 271-281
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Hausdorff Metric ◽  
Density Theorem ◽  
Mosco Convergence ◽  
Convexity Theorem ◽  
Convergence Theorems ◽  
Convergence Of Sets ◽  
Differentiability Properties

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

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