Convex functional

A functional, defined on a convex subset of a linear vector space, the supergraph of which is a convex set. A functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262501.png" /> which does not assume the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262502.png" /> on a convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262503.png" /> is convex on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262504.png" /> if and only if the inequality

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262505.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262506.png" />

is satisfied. If the inequality sign is reversed, the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262507.png" /> is called concave. Operations which convert a convex functional into a convex functional include addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262508.png" />, multiplication by a positive number, taking the upper bound

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c0262509.png" />

and the infimal convolution

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c02625010.png" />

A convex functional bounded from above in a neighbourhood of some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c02625011.png" /> is continuous at that point. If a convex functional is finite at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c02625012.png" />, it has a (finite or infinite) derivative in any direction at that point. Closed convex functionals (i.e. functionals with convex and closed supergraphs) in locally convex linear topological spaces may be described in a dual way: Such a functional is the least upper bound of the affine functions which it dominates. This duality makes it possible to relate to each convex functional a dual object, the conjugate functional

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026250/c02625013.png" />

Properties of convex functionals, operations on such functionals, and the relationship between a convex functional and its conjugate, are studied in convex analysis.

References

[1]	Z.W. Birnbaum, W. Orlicz, "Ueber die Verallgemeinerung des Begriffes der zueinander konjugierter Potenzen" Studia Math. , 3 (1931) pp. 1–67 MR Template:ZBL
[2]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) MR2591337 MR1601748 MR1472649 MR1478445 MR1234454 MR1146141 MR1136994 MR0944909 MR0197653 MR0046395 MR0083530 MR0008854 MR1563622 MR1545539 MR1545530 Template:ZBL Template:ZBL
[3]	M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian) MR Template:ZBL
[4]	W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77 MR0028365 Template:ZBL
[5]	R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) MR0274683 Template:ZBL