Shape of polar set

Question

Let $K$ be a subset of $R^n$, which contains the origin $\theta$ , maybe , it is needed that it is not very strange . The polar set of $K$ is $$ K^0=\{x\in R^n : \langle x,y \rangle \le 1 ~~~\forall y \in K \} $$ I try to image the shape of $K^0$ , but only when $K$ is sphere with origin as centre I can image the $K^0$ (it is sphere too. Just maybe the radial is not same.). Whether there is easy way to image the shape of $K^0$ after $K$ is given ?

Answer 1

If $K$ is the convex hull of a finite number of points $\{p_i\}_{i=1}^m$, $K^\circ$ can be (trivially) written as intersection of halfspaces: $$K^\circ = \bigcap_{i=1}^m \{x \in \mathbb{R}^n \mid x^\top p_i \le 1\}.$$

This helps to get some feeling for small numbers $m$.