6
$\begingroup$

It is very common to indicate with $\Gamma_0(A)$ the set of lower semicontinuous convex functions from $A$ to $(-\infty,+\infty]$ with nonempty domain. An example of usage of this notation can be found in "Proximal Splitting Methods in Signal Processing" by Combettes and Pesquet.

What is the role of the subscript $0$ in the short notation?

Improve this question
$\endgroup$
0

2 Answers 2

Reset to default
11
$\begingroup$

I think Carlo Beenakker digged up the right reference for the notation of the set, but I think more can be said.

First, there is some meaning for the subscript $0$ which can be found in the same paper of Moreau a little later:

excerpt of the paper

So $\Gamma(H)$ are all functions that are pointwise suprema of continuous affine functions. This is a neat space since it's elements are build from very simple objects, but is also quite general as it includes all convex and lower semi-continuous functions. However, it includes two not-so-nice functions, namely $f(x)\equiv \infty$ and $f(x)\equiv -\infty$. Both are not really needed and complicate the theory a bit, hence $\Gamma_0(H)$ is introduced which is just $\Gamma(H)$ without these two functions (and adding a subscript $0$ to specialize a set seems pretty common).

Also, one can say a bit more about the letter $\Gamma$. I don't think its origin has something to do with projections. I checked Fenchel's lecture notes on convexity from 1951. There he did not use convex functions that were allowed to assume the value $+\infty$, but always includes the domain of definition into the description of the convex functions, i.e. he wrote $[C,f]$ to denote a convex function $f:C\to{}]{-\infty},+\infty[$.

In this lecture notes he describes dualities between convex sets (Chapter II, Sec. 8) and also between functions (Chapter III, Sec. 5). For the duality of sets he used $C$ for sets of points and $\Gamma$ for sets of hyperplanes and then describes the duality of the two descriptions of convexity by either 1) the points in the set or 2) by the hyperplanes that separate the set from outer points. In III.8 where he describes Fenchel duality he used $[C,f]$ for "convex $f$ defined on $C$" and denotes the dual function as $[\Gamma,\varphi] = [C,f]^*$. Here $\Gamma$ somehow describes (a part of) the affine functions minorizing $f$ which may be the motivation for Moreau to use $\Gamma$ for the set.

Improve this answer
$\endgroup$
4
  • $\begingroup$ Thank you for your detailed answer. Could we therefore say that $\Gamma_0(H)$ is the set of the functions belonging to $\Gamma(H)$ that admit the horizontal (zero) hyperplane as supporting hyperplane? $\endgroup$
    – MMFF
    Commented Feb 22, 2017 at 15:31
  • $\begingroup$ You're welcome. The answer to your question is no, since affine functions are also in $\Gamma_0(H)$. I don't think that you get a more handy description as "convex, proper and lsc" or "suprema of affine function, but not constant $\pm\infty$. $\endgroup$
    – Dirk
    Commented Feb 22, 2017 at 15:38
  • 1
    $\begingroup$ Oh, right. I should remember more often my professor's motto: "look at the boundary". Than you again! $\endgroup$
    – MMFF
    Commented Feb 22, 2017 at 16:10
  • $\begingroup$ I suspect $C$ and $\Gamma$ are used together because $C$ is the "Latin version" of $\Gamma$, in some sense (they are both third in alphabetical order, and apparently $C$ was derived from $\Gamma$ by the Etruscans). $\endgroup$
    – Robert Furber
    Commented Mar 29, 2019 at 12:55
7
$\begingroup$

It seems the notation $\Gamma_0$ was introduced by Jean-Jacques Moreau in Proximité et dualité dans un espace Hilbertien (1965), as a generalisation of the notion of projection onto a convex domain (see screenshot). No explanation is given for the subscript $0$, but the connotation with a projection seems natural.

Improve this answer
$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .