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.