It is common to use: $$\| \sum_{i=1}^n \bar X_i \| \le \sum_{i=1}^n \|\bar X_i \|. $$
Now for 2 dimensions, I know that:
$$ \|\bar A + \bar B \|^2 = (\bar A + \bar B)\cdot(\bar A + \bar B)$$ $$ = \bar A \cdot \bar A + 2 \bar A \cdot \bar B + \bar B \cdot \bar B $$ $$ = \|\bar A\|^2 + \|\bar B\|^2 + 2 \bar A \cdot \bar B $$
and since $\bar A \cdot \bar B = \|\bar A\| \|\bar B\|\ cos\ \theta \ge -\|\bar A\| \|\bar B\|:$
$$ \|\bar A + \bar B\|^2 \ge \|\bar A\|^2 + \|\bar B\|^2 - 2\|\bar A\|\|\bar B|| $$ $$ = \left(\|\bar A\| - \|\bar B\|\right)^2 $$ $$ \rightarrow \|\bar A + \bar B\| \ge \|\bar A\| - \|\bar B\|, $$
which gives a lower bound for $ \|\bar A + \bar B\|. $
I have tried to generalize this approach for more dimensions, but the best I got was:
$$ \Vert \bar A + \bar B + \bar C \Vert^2 = \left( \bar A + \bar B + \bar C \right) \cdot \left( \bar A + \bar B + \bar C \right) $$ $$ = \bar A \cdot \bar A + \bar B \cdot \bar B + \bar C \cdot \bar C + 2\left( \bar A \cdot \bar B + \bar B \cdot \bar C + \bar A \cdot \bar C \right) $$ $$ \ge \Vert \bar A \Vert^2 + \Vert \bar B \Vert^2 + \Vert \bar C \Vert^2 - 2 \left( \Vert \bar A \Vert \Vert \bar B \Vert + \Vert \bar B \Vert \Vert \bar C \Vert + \Vert \bar A \Vert \bar C \Vert \right) $$ $$ = \left( \Vert \bar A \Vert^2 - 2\Vert \bar A \Vert \Vert \bar B \Vert + \Vert \bar B \Vert^2 \right) + \Vert \bar C \Vert^2 - 2 \left( \Vert \bar A \Vert \Vert \bar C \Vert + \Vert \bar B \Vert \Vert \bar C \Vert \right) $$ $$ = \left( \Vert \bar A \Vert - \Vert \bar B \Vert \right)^2 + \Vert \bar C \Vert^2 - 2\Vert \bar C \Vert \left( \Vert \bar A \Vert + \Vert \bar B \Vert \right) $$
which doesn't seem very useful.
Is there a lower bound for the magnitude of the sum of $n$ vectors? (other than $0$, of course). How should I go about proving it?
Actually this question came up when I was trying to find a lower bound for $\left|\int_{-\infty}^t x(\tau) \, d\tau\right|.$ I realize I have gone astray from the original problem, but still both questions interest me. I would appreciate any help, and thanks for reading through whoever does.
The natural generalization is $$ \left\|\sum_{i=1}^n X_i\right\| \ge \|X_1\| - \sum_{i=2}^n \|X_i\|$$
which follows easily from the Triangle Inequality.