Decide convexity

Question

Decide whether the following function is convex $$ f^{*}(x)=\sup_{y \in \operatorname{Dom}(f)} \left\{ x^T y-f(y) \right\}, f(y):\mathbb{R}^n \xrightarrow{} \mathbb{R}. $$ Where $$ \operatorname{Dom} (f)=\left\{ y: f(y) < \infty, y \in \mathbb{R}^n \right\}. $$ Can you give me some hint or theorem which is suitable for my problem?

Answer 1

you can use the definition of convexity :

$$\forall (x,y)\in\mathbb{R}^{n}\times \mathbb{R}^{n}, \forall t\in [0,1 ]\\ f^*((1-t)x+ty)\leq (1-t)f^*(x)+tf^*(y)$$

You can just develop the expression of $f^*$ and use the fact that $sup_z\{a(z)+b(z)\}\leq sup_z\{a(z)\}+sup_z\{b(z)\}$