I am reading through this tutorial on DC programming and the author makes a startling claim without proof: If $g$ and $h$ are two lower semi-continuous convex function, then the conjugate function of their difference
$$(g-h)^*(y) \equiv \sup\{x'y - (g-h)(x)\} = h^*(y) - g^*(y) $$ where $h^*$ and $g^*$ are their conjugates respectively $g^*(y) \equiv \sup\{x'y - g(x)\}$, etc.
I'm not able to convince myself that this statement is true. Any ideas / suggestions on how its proof might go?
The claim is false. In general, the conjugate of $g-h$ is not even finite: e.g., $g\equiv 0$, $h(x)=x^2$.
And when it is finite, the equality has no reason to hold. Let $g(x)=Ax^2$ and $h(x)=Bx^2$ with $A>B>0$. Then $g^*(x)=\frac{1}{4A}x^2$, $h^*(x)=\frac{1}{4B}x^2$, and $(g-h)^*(x)=\frac{1}{4(A-B)}x^2$. Obviously, we should not expect $1/(A-B)$ to be equal to $1/B-1/A$.
The analogous statements for sums and scalar multiples are also false. The Legendre transform is very nonlinear.