Assuming $D\subset R^{n+1}$ is smooth and strictly convex region, one can construct an embedding $\varphi$ of $S^n$ to the boundary $\partial D$ of $D$ which associates to each direction $z\in S^n$ the point of $\partial D$ with normal $z$, where $S^n$ is unit sphere. This is given by $$ \varphi(z)= s(z)z+\nabla s(z) \tag{1} $$ where $s$ is support function of $D$, $$ s(z)=\sup_{y\in D} \langle y, z\rangle $$
and $\nabla s$ is the gradient vector of $s$ with respect to the standard metric $g$ on $S^n$.
The above statement is from Andrews, Ben, Entropy estimates for evolving hypersurfaces, Commun. Anal. Geom. 2, No. 1, 53-64 (1994). ZBL0839.53049.
I want to show the $\varphi(z)$ is the point of $\partial D$ with normal $z$, but don't know how to begin. There are two things need to be proved:
1, How to show $\varphi(z)\in \partial D$ ?
2, How to show the normal of $\varphi(z)$ is $z$?
Besides, assuming $\{e_1, ...,e_n\}$ is a local basis of $S^n$, I understand the gradient vector $\nabla s$ is $$ \nabla s = g^{ij}(\nabla _{e_i} s) e_j $$