Given a family of $n$ vectors, each of distance $1$ from the span of the rest. Is the norm of their sum at least $\sqrt{n}$?

Question

Let $v_1,\dots,v_n$ be vectors in $\mathbb{R}^n$, not necessarily linearly independent. For each $v_i$ let $v_i'$ denote the projection of $v_i$ into the orthogonal complement of $\mathrm{span}\{v_1,\dots,v_{i-1},v_{i+1},\dots,v_n\}$. Is it true that

$$\|\sum_{i=1}^n v_i\|_2^2 \ge \sum_{i=1}^n \|v_i'\|_2^2$$

This is trivial when all vectors are orthogonal, but now I don't know if it holds in general.

Answer 1

No; take $(1,0),(-1,1)\in\Bbb{R}^2$.