is there a sufficient and necessary condition for Minkowski inequality when $p= \infty$?
i.e. When does the equality $\|f\|_\infty+ \|g\|_\infty =\|f+g\|_\infty $ hold?
The norm here is the essential supremum
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If $|f|_{\infty}$ the essential supremum of $|f|$ or just $\sup|f|$ ? – Jack D'Aurizio Jan 10 '15 at 11:14
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Oh, I am sorry, it's essential supremum – No One Jan 10 '15 at 17:30
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This has been asked before, with no answer given. I'll argue that there is no answer that is not tautological: there are so many pairs $(f,g)$ for which the equality holds, that the property $\|f+g\|_\infty=\|f\|_\infty+\|g\|_\infty$ is the simplest description of all such pairs.
For simplicity, scale the functions so that $\|f\|_\infty=1=\|g\|_\infty$. Introduce the vector-valued function $H(x)=(f(x),g(x))$ and let $R$ denote its essential range: the smallest closed subset of $\mathbb R^2$ such that the set $\{x:H(x)\notin R\}$ has measure zero. So, $R$ is a compact subset of $[-1,1]^2$.
The equality $\|f+g\|_\infty$ holds if and only if $R\cap \{(1,1),(-1,-1)\}$ is nonempty. Indeed, the function $(x,y)\mapsto |x+y|$ attains its extreme values on $R$, so in order for its supremum to be $2$, the set $R$ has to contain a point where the value $2$ is attained.
How to describe the compact subsets of $[-1,1]^2$ that contain either $(1,1)$ or $(-1,-1)$? Well, they are what they are.
