I am trying to find necessary and sufficient conditions for the triangle equality to be satisfied:
$$|x_1 + x_2 + \cdots + x_n| = |x_1| + |x_2| + \cdots + |x_n|$$
where $x_i$ are vectors in $\mathbb R^n$.
I am guessing the condition is analogous to the $|x+y|\le|x|+|y|$ case, namely that $x_1, x_2, \ldots x_n$ are a linearly dependent set. I can prove this is necessary, but not that it is sufficient, which it may well not be.
1 Answer
Suppose that none of $x_1,x_2,...,x_{n+1}$ is zero, observe that when $n=1$ $$|x_1+x_2|=|x_1|+|x_2| $$ if and only if $\exists c_1>0$ such that $x_2=c_1x_1$.
By triangle inequality $|x_1+...+x_n+x_{n+1}|\le |x_1+...+x_n|+|x_{n+1}|$ so if $$|x_1+...+x_n+x_{n+1}|= |x_1|+...+|x_n|+|x_{n+1}|$$ ,we'd have $|x_1|+...+|x_n|\le|x_1+...+x_n|$ which implies that $$|x_1|+...+|x_n|=|x_1+...+x_n|$$
Proceed inductively, assume that $$|x_1+...+x_n|=|x_1|+...+|x_n| $$ if and only if $\exists c_1,...,c_{n-1}>0$ such that $x_{i+1}=c_ix_1$ whenever $i=1,2,...,n-1$. If we have $$|x_1+...+x_n+x_{n+1}|=|x_1|+...+|x_n|+|x_{n+1}| $$ then $$|x_1+...+x_n+x_{n+1}|=|x_1+...+x_n|+|x_{n+1}| $$ so $\exists c^*_n>0$ such that $x_{n+1}=c^*_n(x_1+...+x_n)=c^*_n(1+c_1+...+c_{n-1})x_1$. By letting $c_n=c^*_n(1+c_1+...+c_{n-1})$, we arrive the conclusion that
$|x_1+...+x_n+x_{n+1}|=|x_1|+...+|x_n|+|x_{n+1}|$
if and only if $\exists c_1,...,c_n>0$ such that $x_{i+1}=c_ix_1$.
The case where some of $x_i=0$ can be tackled analogously with some caution.
