Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by $$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$ acts via symplectomorphisms (where $x\in \mathbb{R}^2$, $\xi\in T^∗_𝑥\mathbb{R}^2$, and $\varphi_g$ is the map which acts as $x↦g\cdot x$). If the action on the base preserves a convex region $K\subset \mathbb{R}^2$, containing the origin and centrally symmetric ($x\in K$ iff $-x\in K$), is it true that the action on the cotangent space preserves $K\times K^{\circ}$ where $K^{\circ}$ is the dual convex body of $K$.

This is certainly true for $K$ being the ball, but I have not been able to draw either a positive or a negative answer in general. I would very much appreciate any help or suggestion.

Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by $$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$ acts via symplectomorphisms (where $x\in \mathbb{R}^2$, $\xi\in T^∗_𝑥\mathbb{R}^2$, and $\varphi_g$ is the map which acts as $x↦g\cdot x$). If the action on the base preserves a convex region $K\subset \mathbb{R}^2$, containing the origin and centrally symmetric ($x\in K$ iff $-x\in K$), is it true that the action on the cotangent space preserves $K\times K^{\circ}$ where $K^{\circ}$ is the dual convex body of $K$, i.e. $$K^{\circ}:=\{y\in\mathbb{R}^2\mid \langle x,y\rangle\leq 1, \forall x \in K\}.$$

This is certainly true for $K$ being the ball, but I have not been able to draw either a positive or a negative answer in general. I would very much appreciate any help or suggestion.