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Why it is important to know the projection onto epigraph of a function?

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Why, in general, someone should be interested in finding projections onto epigraph(f), where $f:X\to\mathbb{R}$ is a given function? ($X$ Hilbert space).

I heard about that in problems related to convex optimization, but I don't understand why this issue is so important.

I hope someone could help me. Thank you in advance!

${\bf EDIT:}$ I refer to these notes: https://gubner.ece.wisc.edu/notes/ConvexityNotes.pdf

At the end of page 5, the author says that: "Since we will be interested in projections onto epi f , itis important to have conditions on f under which epi($f$) will be nonempty, closed, and convex." I don't understand why he is interested in that.


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