Infimal convolution $g^\star = f_1^\star + f_2^\star$ Let $f_1$ and $f_2$ be convex functions on $R^n$. Their infimal convolution $g = f_1 \diamond f_2$ is defined as 
$$
g(x) = \inf \{f_1(x_1) + f_2(x_2) \mid x_1 + x_2 = x\}.
$$
Prove that $g^\star = f_1^\star + f_2^\star$.

I finally found a solution to the problem
$$
g*(y) = \underset{x}{\sup}\; \{x^Ty - g(x)\} 
$$
As we know 
$$
g(y) = \underset{x_1 + x_2 = x}{\inf}\; \{f_1(x_1) + f_2(x_2)\} 
$$
Now we have
$$
g*(y) = \underset{x}{\sup}\; \{x^Ty - \underset{x_1 + x_2 = x}{\inf}\; \{f_1(x_1) + f_2(x_2)\} \}  
$$
$$
= \underset{x}{\sup}\; \{x^Ty + \underset{x_1 + x_2 = x}{\sup}\; \{-f_1(x_1) - f_2(x_2)\} \}  
$$
$$
= \underset{x}{\sup}\underset{x_1 + x_2 = x}{\sup}\; \{x^Ty -f_1(x_1) - f_2(x_2)\} \}  
$$
$$
= \underset{x_1,x_2}{\sup}\; \{x_1^Ty + x_2^Ty -f_1(x_1) - f_2(x_2)\} \}  
$$
$$
=  f_1^\star(x_1) + f_2^\star(x_2)
$$
