TITLE: The convex hull of a manifold whose cobordism class is trivial
QUESTION [6 upvotes]: Let $M$ be a compact orientable  $n$ dimensional  manifold. Assume that $M$ has trivial cobordism class. 
Is there an embedding  of $M$ in some Euclidean space $\mathbb{R}^m$ such that the  convex hull of $M$ is a $n+1$ dimensional manifold whose boundary is $M$?
Here the image  of  $M$ under the   embedding is denoted again by $M$.
Note: One can pose the same question in the following geometric manner:
Let $M$ be a compact Riemannian manifold with trivial cobordism class. Is  there an isometric embedding of  $M$ in some Euclidean space such that the convex hull of  $M$ is  a manifold whose boundary is $M$?

REPLY [12 votes]: Implicit in the other responses is the fact that if $M$ bounds a convex manifold $W$, then $W$ is contractible and so M has the homology of a sphere. So any null-cobordant manifold that is not a homology sphere is a counterexample. Eg take $M = X \# -X$ where $X$ is any orientable manifold with non-trivial homology.