Let $x \in \mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $\mathbb{R}$.

Is the following property true? $$ \max_{x\in\mathbb{R}^n} \{ f(x) + g(x) \} = \max_{x\in\mathbb{R}^n} f(x) + \max_{x\in\mathbb{R}^n} g(x) $$ Of course, for arbitrary functions this is $\leq$ instead of $=$, but I need this property in a larger proof and I am not sure if it is true or false.

Could anyone verify, and possibly sketch a small proof?

Greatly appreciated.

  • 1
    $\begingroup$ Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist? $\endgroup$ – Hermione Dec 8 '18 at 11:44

This cannot be true since the maximum of an affine function $f$ on $\Bbb R^n$ is always $+\infty$ unless it's a constant function.

  • $\begingroup$ This is actually very helpful. My specific case is $\max_\nu \{ \langle b,\nu \rangle + \langle \nu, u \rangle + \Vert u \Vert \}$. So this is only equal to $\max_\nu \langle b,\nu \rangle + \max_\nu \{ \langle \nu, u \rangle + \Vert u \Vert \}$ if I set the condition that $\langle \nu, u \rangle = - \Vert u \Vert$, a constant. Correct? $\endgroup$ – ex.nihil Dec 8 '18 at 11:59

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