A $n-1$ dimensional submanifold $N\subset \mathbb{R}^n$ is called a convex submanifold if for every $x\in N$ ,ther is a neighborhood $W$ of $x$ in $N$ such that $W$ entirly lies at one side of $T_x N$. A (local) diffeomorphism $\phi$ on $\mathbb{R}^n$ is called a convex diffeomorphism if $\phi$ and its inverse preserves the convexity of all codimension $1$ submanifolds. For example every affine linear isomorphism is a convex diffeomorphism but the diffeomorphism $\phi(x,y)=(x,y-x^2+x^3)$ is not a convex diffeomorphism because it maps the convex curve $y=x^2$ to non convex curve $y=x^3$.

1.In the above definition, is the word "its inverse", redundant?

- Is there a well known description of the group of all smooth convex diffeomorphisms of $\mathbb{R}^n$?
3.Does every manifold admit an atlas whose all transition maps are convex diffeomorphism?

If the answer to the later question is affirmative, then we can define the concept of convexity for any codimension $1$ submanifold of an arbitrary manifold $M$. In particular we can speak of "convex foliation" of a manifold $M$, a codimension $1$ foliation of $M$ whose all leaves are convex submanifold. In this case the next question would be the following:

4.Is there a manifold which admit a codimension $1$ foliation but does not admit any convex foliation?In particular does $S^3$ admit a convex foliation?

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