Under what condition sum of the norm squared is less than of square of the norm of average

Question

Let $x_1, x_2, \dots, x_N \in \mathbb{R}^n$ be some given vectors such that $\sum_{i=1}^N x_i \neq 0$. Is there a constant $c>0$ such that the following holds? $$ \sum_{i=1}^N\| x_i\|^2 \leq c\|\sum_{i=1}^N x_i\|^2 $$ If so, what that $c$ would be?

Answer 1

No, the claim is false. Take for example $$x_1=e_1, \hspace{10pt} x_2=(-1+\frac{1}{k})e_1$$ for $k$ arbitrarily great (and $x_h=0$ for every $h \geq 3$).