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

  In general $X$ is a convex subset of $Y$ if any two points in $X$ is joined by distance-minimizing geodesic which lies in $X$


 Question 1) Does the boundary of $m$-dimensional bounded convex set has dimension $m-1$ ?


 Question 2) Is the following opinion is right ?

  $(\ast)$ My thought : Let $m\geq 2$. A $(m-1)$-dimensional boundary of a $m$-dimensional bounded closed convex set $X$ is smooth 
  except some $(m-2)$-dimensional set. 

   


  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.