Equality condition in Minkowski's inequality for $L^{\infty}$
I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case where $1<p<\infty$ but not when $p=\infty$ so I am hoping to get some ideas or for someone to point me to a source where this is discussed.
I will list a couple of observations I made while working this out (though I'm not sure whether I'm right with these):
If $\mu(\{x:|f(x)|\geq\|f+g\|_{\infty}-\|g\|_{\infty}\})=0$ (or with $f$ and $g$ interchanged), then I have the reverse inequality.
If I pick $a,b$ such that $\|f\|_{\infty}\leq a<\|f\|_{\infty}+\varepsilon$, $\|g\|_{\infty}\leq b<\|g\|_{\infty}+\varepsilon$, $\mu(\{x:|f(x)|>a\})=0$, $\mu(\{x:|g(x)|>b\})=0$, and for all $c<a+b$ I have $\mu(\{x:|f(x)+g(x)|>c\})>0$, then I also have the reverse inequality.
There is no simple condition for equality here. For example if $f=\chi_{(0,3)},g=\chi_{(1,2)}$ then $||f+g||_\infty =||f||_\infty +||g||_\infty$.But f and g are not multiples of each other. I hope this example convinces you that one cannot write down simple N & S conditions for equality in Minkowski's inequality.