I recently asked this question.

I think, if the following were true, then I would solve my problem.

Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex set.

$E$ is a submanifold of $\mathbb R^n$. For all small enough $\epsilon>0$ I can thus pick a tubular neighbourhood around it.

Is there any way to show that I can pick such a neighbourhood with a boundary which is nowhere piece-wise linear and such that the neighbourhood (together with $E$) is convex?

All help greatly apreciated.