How is the inequality constraint $\mbox{Tr}(W) \geq c$ convex?

Question

In one paper, I found that the following inequality constraint $$\mbox{Tr}(W) \geq c$$ where $W$ is a symmetric positive semidefinite matrix variable and $c$ is a constant, is convex.

In my understanding $\mbox{Tr}(W)$ is a convex function and, therefore, the above constraint should not define a convex set. Is there something wrong with my understanding?

Answer 1

Suppose $$Tr(X) \ge c$$

and $$Tr(Y) \ge c$$

For $\alpha \in (0,1)$, we have $$Tr(\alpha X + (1-\alpha) Y)=\alpha Tr(X)+(1-\alpha)Tr(Y) \ge c$$

Hence it is convex.

Note that trace is a linear function.

Answer 2

$\text{Tr}(W)$ is a linear function, so the constraint is convex.