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I'm confused with the definition of the closed proper convex functions when reading the paper

https://people.orie.cornell.edu/aslewis/publications/00-dykstras.pdf

It appears that, when a function $f$ is said to be closed, convex, proper, Legendre, co-finite and very strictly convex, it must be defined on the whose space $\mathbb R^J$. The function $f(x):=x\log(x)-x$ given in Example 2.11 is also called closed, convex, proper, Legendre, co-finite and very strictly convex. However it is only defined on $\mathbb R_+$ instead of $\mathbb R$.

Can anyone explain this? In particular, for the main results Theorem 3.2 and 4.3, do they hold for the above function $f$?

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  • $\begingroup$ I think what is meant there is that $f=\infty$ on $(-\infty,0)$ and $f(0)=0$. $\endgroup$
    – Iosif Pinelis
    Commented Nov 30 at 22:16
  • $\begingroup$ @IosifPinelis You are right. After a check $f$ can be valued in $(-\infty,\infty]$. So this extension is adopted without precision. I was not aware of that $\endgroup$
    – Fawen90
    Commented Dec 1 at 9:17

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