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 $M$?



The convex hull of a manifold whose cobordism class is trivial
There are exotic spheres (which are null cobordant) which do not bound a parallelisable manifold. Since the convex hull is contractible, it would be parallelisable if it were a manifold, so these guys do not admit embeddings like you want.


