Polar of a set is defined as:
$C^0 = \{ y\in \mathbb{R}^n | y^Tx \leq 1, \forall x \in C\}$
Now from wikipedia, the intuitive idea of polar of a cone is easily understandable. But how can it be shown that $C^0$ is convex, even when $C$ is not? Any hint or explanation is highly appreciated. Thanks.
$C^0$ can be written as the intersection of closed half-spaces: $$ C^0 = \bigcap_{x\in C} \{ y\in \mathbb R^n: x^Ty\le 1\}, $$ hence $C^0$ is convex.
If $C$ is cone then $$ C^0 = \{ y\in \mathbb R^n: x^Ty\le 0 \ \forall x\in C\}, $$ which is sometimes called the dual cone. If $C$ is a linear subspace then $C^0 = C^\perp$.