Definition
A convex curve/surface is a simple curve/surface in the Euclidean plane/space which lies completely on one side of each and every one of its tangent lines/planes.
So more generally we give the following definition
Definition
A $(n-1)$-manifold of the $n$-dimensional euclidean space is convex if it lies completely on one side of each and every one of its tangent space.
So with the previous definition I ask to me if a $(n-1)$-manifold $M$ disconnects the euclidean space into two connected part. First of all I observed that the intersection $U$ of the positive tangent spaces is not empty because $M$ is there contained trivially and thus being any positive tangent space connected surely $U$ is even connected so that the statement follows showing that the complement of $U$ is connected too but unfortunately I do not be able to do it and thus I ask to prove it. Moreover is my argument about the connectdness of $U$ correct? So could someone help me?