Let $\mathbf{S}$ be the unit Euclidean sphere in $\mathbf{R}^n$. I write $u \bullet v$ for the scalar product of two vectors and $A \sim B$ for the set-theoretic difference of sets.
Assume $g : \mathbf{S} \to \mathbf{S}$ is a smooth diffeomorphism and $h : \mathbf{R}^n \sim \{0\} \to \mathbf{S}$ is its $0$-homogeneous extension, i.e., $h(x) = g(x/|x|)$ for $x \in \mathbf{R}^n \sim \{0\}$. Assume $h$ is monotone in the sense that $$ \bigl[ \mathbf{R} \ni t \mapsto h(x + tv) \bullet v \bigr] \quad \text{is non-decreasing for each $x,v \in \mathbf{R}^n \sim \{0\}$} \,. $$
What else do we need to assume on $g$ to ensure the existence of a bounded convex body $C$ such that for each $v \in \mathbf{S}$ the exterior normal vector to $C$ at the point $\partial C \cap \{tv : t > 0 \}$ equals $g(v)$?
Equivalently:
What more do we need to assume on $g$ to ensure the existence of a convex positively $1$-homogeneous function $\phi : \mathbf{R}^n \sim \{0\} \to \mathbf{R} \sim \{0\}$ (i.e. a possibly non-symmetric norm) such that $\operatorname{grad} \phi(v) \in \{ t g(v) : t > 0 \}$ for $v \in \mathbf{S}$?
