We have set $A \subseteq \mathbb{R}^n $ and define $A^*$ as $$ A^* := \{s \in \mathbb{R}^n : x^Ts \geq 0, \forall x \in A \}$$
Suppose $A^*$ is nonempty. What are the conditions needed for $A = (A^*)^*$?
(Question actually asks to prove (or disprove) that $A$ is a closed convex cone.)