  We consider the problem $\inf\big\{ \int\!\!\int_\Omega (1 + |\nabla
u(x,y)|^2)^{-1} dx dy : \text{ the function } u : \Omega \to \mathbb{R} \text{
is concave and } 0 \le u(x,y) \le M \text{ for all } (x,y) \in \Omega =\{
(x,y): x^2 + y^2 \le 1 \} \, \big\}$ (Newton's problem) and its
generalizations. In the paper \cite{BrFK} it is proved that if a solution $u$
is $C^2$ in an open set $\mathcal{U} \subset \Omega$ then $\det D^2u = 0$ in
$\mathcal{U}$. It follows that graph$(u)\rfloor_\mathcal{U}$ does not contain
extreme points of the subgraph of $u$.
  In this paper we prove a somewhat stronger result. Namely, there exists a
solution $u$ possessing the following property. If $u$ is $C^1$ in an open set
$\mathcal{U} \subset \Omega$ then graph$(u\rfloor_\mathcal{U})$ does not
contain extreme points of the convex body $C_u = \{ (x,y,z) :\, (x,y) \in
\Omega,\ 0 \le z \le u(x,y) \}$. As a consequence, we have $C_u = \text{\rm
Conv} (\overline{\text{\rm Sing$C_u$}})$, where Sing$C_u$ denotes the set of
singular points of $\partial C_u$. We prove a similar result for a generalized
Newton's problem.
