Theorem 5.1 gives local $W^{2,1}$ regularity on $\Omega-\Sigma$. By Theorem 1.1, for any $\eta>0$ we can cover $\Sigma$ by balls $\{B_{r_{i}}(x_{i})\}$ such that

$$\sum_{i=1}^{\infty}r_{i}^{n-1}<\eta.$$

Let $A=\cup_{i=1}^{\infty}B_{r_{i}}(x_{i})$. Since $u$ is a convex function, the second derivatives are controlled by $\Delta u$. It follows that

	$\displaystyle\int_{A}\|D^{2}u\|\,dx$	$\displaystyle\leq\int_{A}\Delta u\,dx$
		$\displaystyle\leq\sum_{i=1}^{\infty}\int_{\partial B_{r_{i}}}u_{\nu}\,ds$
		$\displaystyle\leq C\sum_{i=1}^{\infty}r_{i}^{n-1}$
		$\displaystyle\leq C\eta,$

where $C$ is the Lipschitz constant of $u$. This shows that the second derivatives cannot concentrate on $\Sigma$. ∎