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A function is called subadditive such that $f(x+y)\le f(x)+f(y)$ holds for any $x$, $y$ in the domain of $f$. (Let us say that, for example, the domain is some subset of $\mathbb R$ closed under addition. Sometimes a more general case of functions on $\mathbb R^n$ is studied.) Subadditive functions have many useful and interesting properties.

I am interested in the functions fulfilling stronger inequality $f(x+y) \le \max\{f(x),f(y)\}.$

  • Is there a name for functions with the property?
  • Is there some book or survey paper describing basic properties of such functions, which you would recommend? Or are there at least some papers studying such functions?
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    Sounds similar to [Quasiconvex](http://en.wikipedia.org/wiki/Quasiconvex_function). Who's next with a Wikipedia link? :) But seriously, the property simply says that every sublevel set $\{f(x)\le M\}$ is closed under addition. This is not the same as being convex, of course. Such a set will have to be either unbounded or $\{0\}$. Indeed, the function will look like a norm.2012-06-22

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