Let $f\colon C\to\mathbb{R}$ be a concave function of Legendre type defined on an open set $C\subset N_{\mathbb{R}}$ and let $D=\nabla f(C)\subset M_{\mathbb{R}}$ be the image of the gradient map. Then

1. (1)

   $D=\operatorname{stab}(f)^{\circ}$;

2. (2)

   $f^{\vee}|_{D}$ is a concave function of Legendre type;

3. (3)

   $\nabla f\colon C\to D$ is a homeomorphism and $(\nabla f)^{-1}=\nabla f^{\vee}$;

4. (4)

   for all $x\in D$ we have $f^{\vee}(x)=\langle x,(\nabla f)^{-1}(x)\rangle-f((\nabla f)^{-1}(x))$.