What is the name of this inequality?

Question

Let $X_1, \ldots, X_n$ be a collection of nonnegative random variables. Then $$ \max_i X_i \le \epsilon/2 + \sum_i X_i \mathbb{1}_{\{X_i > \epsilon/2\}}. $$ Does this inequality have a name? What is it?

Answer 1

I don't think this has a name. But it is a combination of some simple [but also unnamed] results.

Under the event $\max_i X_i \le \epsilon / 2$, then the inequality holds automatically.

So we may assume we are in the event $\max_i X_i > \epsilon / 2$. We then have $\max_i X_i \le \sum_i X_i \mathbb{1}_{X_i > \epsilon / 2}$ because the right-hand side equals $\max_i X_i$ plus some other nonegative addends.