Upper bound on $\|x\|^2 + \|y\|^2$

Question

I was wondering if there exists a bound in the literature, of the following form

$$\|x\|^2 + \|y\|^2 \leq c \|x+y\|^2$$ where $c > 0$.

I know that if $\langle x, y\rangle \geq 0$, then a possible choice is $c = 1$, but can we say something in the general case.

Answer 1

Just take $y=-x$ to see that you cannot have such an inequality.