Given a centrally symmetric convex body $K$ in the plane (with smooth boundary), it is easy to see that there exists a norm function $g:\mathbb{R}^2\to \mathbb{R}_{\geq 0}$ for which $K$ is the unit ball.

For the polar body $K^{\circ}$ we also have a norm function $h:\mathbb{R}^2\to \mathbb{R}_{\geq 0}$ for which $K^{\circ}$ is the unit ball. I have the following question:

Is $x\times \nabla g(x)=y\times \nabla h(y)$ whenever $(x,y)$ is s.t. $g(x)=h(y)$? Where $\nabla$ denotes the gradient of the function (it makes sense outside the origin in this case) and $x\times y:=x_1y_2-x_2y_1$.

This is certainly true for the K the unit ball in the $\ell^2$-norm.

Oddly enough, if we change $\times$ by the usual inner product this assertion follows from Euler's theorem on homogeneous functions.

Thanks in advance.

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