Convex conjugate of an invertible linear transform In papers I read sometimes that the convex conjugate of 
$$f(Lx) : L \mbox{ invertible}$$
 is
$$ f^*({L^{-1}}^T y)$$
The convex conjugate is defined as 
$$ f^*: {\mathbb{R}^N}^* \rightarrow \mathbb{R}: y \mapsto \sup_{x \in \mathbb{R}^N} \ \langle x, y \rangle - f(x) $$

I tried to comprehend how to come up with the conjugate but I am failing. Can anybody help me up? Thanks
 A: Let $A :=L^{-1}$ and introduce the auxiliary function $g = f \circ L: x \mapsto f(Lx)$.
Now, for any $y \in \mathbb R^N$, one has
$$
\begin{split}
g^*(y) &:= \sup_{x \in \mathbb R^N}\langle x, y\rangle - g(y) = \sup_{x \in \mathbb R^N}\langle x, y\rangle - f(Lx)\\
&= \sup_{z \in \mathbb R^N}\langle Az, y\rangle - f(z)\,(\text{via change of variable }z = Lx\\
&=\sup_{z \in \mathbb R^N}\langle z, A^\top y\rangle - f(z) =: f^*(A^\top y) = f^\star(L^{-\top}y),
\end{split}
$$
as claimed. $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\Box$
An application
If $\|\cdot\|$ is a norm on $\mathbb R^N$ and $\|y\|_\star := \max\{\langle x,y\rangle  \mid x \in \mathbb R^N,\,\|x\| \le 1\}$ is its dual norm, then the convex conjugate of the induced norm $\|\cdot\|_A$ defined by $\|x\|_A := \|Ax\|$ is given by
$$
\|y\|_A^\star = \begin{cases}
0,&\mbox{ if }\|A^{-1} y\|_\star \le 1,\\
\infty,&\mbox{ else.}
\end{cases}
$$
