I'm self-learning Yufei Zhao's "Graph Theorey and Additive Combinatorics", one problem I encounter is the following:
The next exercise can be solved by a neat application of Mantel's theorem.
Exercise 1.1.10. Let $X$ and $Y$ be independent and identically distributed random vectors in $\mathbb{R}^{d}$ according to some arbitrary probability distribution. Prove that $$ \mathbb{P}(|X+Y| \geq 1) \geq \frac{1}{2} \mathbb{P}(|X| \geq 1)^{2} . $$
Mantel's theorem. Every $n$-vertex triangle-free graph has at most $\left\lfloor n^{2} / 4\right\rfloor$ edges. AKA $$ \operatorname{ex}(n, H)=\left\lfloor\frac{n^{2}}{4}\right\rfloor \text {. } $$
I have no clue how this inequality of integral is connected with graph theorey, please at least give me a hint. Proof not using Mantel's theorem is also welcome.
Here is an outline of the strategy.
First, prove that when $x,y,z \in \mathbb R^d$ it is impossible to have $|x|, |y|, |z| \ge 1$ but $|x+y|, |x+z|, |y+z| < 1$. This is our "triangle-free" geometric fact.
Next, consider $n$ points $x_1, \dots, x_n$ in $\mathbb R^d$, of which $|x_1|, \dots, |x_k| \ge 1$ and $|x_{k+1}|, \dots, |x_n| < 1$. Use Mantel's theorem to prove that there are at least $\frac12 k^2$ ordered pairs $(x_i, x_j)$ with $|x_i + x_j| \ge 1$, including pairs $(x_i, x_i)$.
Now, apply this to $n$ points drawn independently from the distribution in the problem. In expectation, we get at least $n(n-1) \Pr[|X+Y| \ge 1]$ pairs $(x_i, x_j)$ with $|x_i + x_j| \ge 1$ and $n \Pr[|X| \ge 1]$ elements $x_i$ with $|x_i| \ge 1$. The application of Mantel's theorem above gives us the inequality we want, with an error term that disappears as $n \to \infty$.