Let $f:X\rightarrow ]-\infty,+\infty]$, let $b\in x$ and let $c\in X$. Find Fenchel conjugates of the functions:

Question

$(i)\;\;\;\; f(x)+\langle c,x\rangle,$

$(ii)\;\;\;\; f(x-c).$

For (i) I'm thinking the formula is $f^*(v)=\sup \langle x,v-c\rangle-f(x)$ and my justification is as follows:

\begin{align} f^*(v)&=\sup \langle x,v\rangle-(f(x)+\langle c,x\rangle)\\ &=\sup \langle x,v\rangle-f(x)-\langle c,x\rangle\\ &=\sup xv -f(x) -xc\\ &=\sup \langle x,v-c\rangle-f(x) \end{align} Does this look correct?

For (ii), intuitively, I want to say $f^*(v)=\sup \langle x,v\rangle-f(x)-(\sup \langle x,v\rangle-f(c))$, but I am not sure how to justify this, so I suspect it i not correct. Any help in the right direction is greatly appreciated.

Ya, you're doing i) perfectly well. You just need to conclude by recognizing the last expression as $f^*(v - c)$ :) — dohmatob, Nov 16 '15 at 22:17
Exercise iii) Compute $g^*$ for $g = \alpha f$, $\alpha \in \mathbb{R}$. — dohmatob, Nov 16 '15 at 22:18

dohmatob · Accepted Answer · 2015-11-16T22:33:10.107

I'll just write down the solution and hope you catch a few tricks from the manipulations :)

i) Define $g = f + \langle c, .\rangle$. For any $x \in X$, we habe \begin{equation} \begin{split} g^*(x) &:= \sup_{z \in X}\langle x, z\rangle - g(z) = \sup_{z \in X}\langle x, z\rangle - \langle c, z\rangle - f(z) = \sup_{z \in X}\langle x - c, z \rangle - f(z) \\ &=: f^*(x - c). \end{split} \end{equation}

ii) Similarly, define $h = f(. - c)$. For any $x\in X$, we have \begin{equation} \begin{split} h^*(x) &:= \sup_{z \in X}\langle x, z\rangle - h(z) = \sup_{z \in X}\langle x, z\rangle - f(z - c) \\ &= \sup_{u \in X}\langle x, u + c\rangle - f(u) (\text{ by the change of variable } u = z - c) \\ & =: \langle x, c\rangle + \sup_{u \in X}\langle x, u\rangle - f(u) =: \langle c, x\rangle + f^*(x). \end{split} \end{equation}

Let $f:X\rightarrow ]-\infty,+\infty]$, let $b\in x$ and let $c\in X$. Find Fenchel conjugates of the functions:

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