I wanto know the smoothness of convex set in ${\bf R}^n$. Recall the following definition. Definition : $X$ is a closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, $tx + (1-t)y \in X$ where $0 < t < 1$.

$(\ast)$ My thought : A boundary of a closed convex set $X$, whose dimension is not 0, is smooth except finite points.

The motivation of this is as follows: In some paper, the Hausdorff measure of convex set in $S^{n-1}(1)$ is considered.

That is, in my thought convex set may be a set of noninteger Hausdorff dimension. Am I right ?

If $\ast$ is right, then why does one consider the Hausdorff measure of convex set ?

Thank you in advance.