Picture below is from the 6th page of Andrews, Ben, Entropy estimates for evolving hypersurfaces, Commun. Anal. Geom. 2, No. 1, 53-64 (1994). ZBL0839.53049.
The $s(z)$ is support function of $D$, and $S^n$ is unit sphere in $R^{n+1}$. I fail to understand some thing.
Firstly, I think the Hessian of $f$ respect to the connection $\overline \nabla$ is $$ Hess_{\overline\nabla} f = (\overline\nabla_i\overline\nabla _j f) $$ I think it can be treated as a linear map from $TS^n$ to $TS^n$ since the matrix can be treated as linear map over vector space. So, I don't know why there is a $\overline g^{~*}$ before $Hess_{\overline\nabla} f$.
Secondly, in (2-5), I guess $d$ is covariant derivative, so the right part of (2-5) can define a map $\overline g^{~*}Hess_{\overline\nabla} f$.
Thirdly, how to show (2-6)? In my view, the Weingarten map is the tangent map induced by Gauss map. It is a map over tangent space, but I don't know how to show (2-6).
PS: I feel hard to understand this paper, and I think I should read some book in preparation for read this paper. But, seemly, the book of Riemannian Geometry don't care about the convex geometry, and the convex geometry is not analytic as in the picture below. Besides, the reference [An1] is not written by English. So, what should I read ?
Thanks for any help.
