scholarly journals The Engulfing Property from a Convex Analysis Viewpoint

2020 ◽  
Vol 187 (2) ◽  
pp. 408-420 ◽  
Author(s):  
Andrea Calogero ◽  
Rita Pini
Keyword(s):  
Simple Proof ◽  
Convex Functions ◽  
Convex Analysis ◽  
Strict Convexity

Abstract In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

Strict convexity, strong ellipticity, and regularity in the calculus of variations

1980 ◽  
Vol 87 (3) ◽  
pp. 501-513 ◽  
Author(s):  
J. M. Ball
Keyword(s):  
Nonlinear Systems ◽  
Calculus Of Variations ◽  
Weak Solutions ◽  
Convex Functions ◽  
Strict Convexity ◽  
Mathematical Tool ◽  
Strong Ellipticity ◽  
Regularity Of Weak Solutions ◽  
Nonlinear Elastostatics

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.

scholarly journals A simple proof of the sum formula

2001 ◽  
Vol 63 (2) ◽  
pp. 337-339 ◽  
Author(s):  
A. Verona ◽  
M. E. Verona
Keyword(s):  
Simple Proof ◽  
Convex Functions ◽  
Short Proof ◽  
Sum Formula

In this note we present a simple, short proof of the sum formula for subdifferentials of convex functions.

Conjugates and Legendre Transforms of Convex Functions

1967 ◽  
Vol 19 ◽  
pp. 200-205 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Strict Convexity ◽  
Legendre Transformation ◽  
Legendre Transform ◽  
Implicit Functions ◽  
The One ◽  
Legendre Transforms ◽  
Effective Domain ◽  
The Relationship

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

scholarly journals A short proof of the separable reduction theorem

2010 ◽  
Vol 43 (3) ◽  
Author(s):  
Jean-Paul Penot
Keyword(s):  
Simple Proof ◽  
Convex Analysis ◽  
Nonsmooth Analysis ◽  
Short Proof ◽  
Reduction Theorem ◽  
Asplund Spaces ◽  
Fréchet Subdifferentials ◽  
Separable Spaces

AbstractWe present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.

scholarly journals A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Sangho Kum ◽  
Yongdo Lim
Keyword(s):  
Convex Functions ◽  
Convex Analysis ◽  
Geometric Mean ◽  
Positive Semidefinite ◽  
Dual Operator ◽  
Positive Semidefinite Matrices ◽  
Essential Properties ◽  
Harmonic Means ◽  
Further Development ◽  
Interesting Generalization

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

scholarly journals Inequalities for certain Classes of Convex Functions

1959 ◽  
Vol 11 (4) ◽  
pp. 231-235 ◽  
Author(s):  
L. Mirsky
Keyword(s):  
Simple Proof ◽  
Convex Functions ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Fan Theorem ◽  
Special Cases ◽  
Gauge Functions ◽  
Ky Fan ◽  
Ky Fan Theorem ◽  
Symmetric Gauge

Making use of properties of doubly-stochastic matrices, I recently gave a simple proof (4) of a theorem of Ky Fan (Theorem 2b below) on symmetric gauge functions. I now propose to show that the same idea can be employed to derive a whole series of results on convex functions ; in particular, certain well-known inequalities of Hardy-Littlewood-Pólya and of Pólya will emerge as specìal cases.

scholarly journals Algebraic hyper-structures associated to convex analysis and applications

Filomat ◽  
2012 ◽  
Vol 26 (1) ◽  
pp. 55-65 ◽  
Author(s):  
Delavar Khalafi ◽  
Bijan Davvaz
Keyword(s):  
Convex Programming ◽  
Convex Functions ◽  
Convex Analysis ◽  
Linear Functions

In this paper, we generalize some concepts of convex analysis such as convex functions and linear functions on hyper-structures. Based on new definitions we obtain some important results in convex programming. A few suitable examples have been given for better understanding.

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