Replacing $\Sigma_{v}$ with

$$\Sigma_{v}^{k}=\{|S_{h,p}^{v}|<C_{p}h^{\frac{n+k}{2}}\},\quad 1\leq k\leq n-1$$

and replacing $1$ with $k$ in the preceding, one obtains that $\mathcal{H}^{n-k}(\Sigma_{v}^{k})=0$. If $\det D^{2}u\geq 1$, such growth happens for $v=u+\frac{1}{2}|x|^{2}$ at points where $u$ agrees with a linear function on a $k$-dimensional subspace. This shows that the Hausdorff dimension of the $k$-dimensional singularities is at most $n-k$. In particular, we recover Lemma 2.3 since for $k\geq\frac{n}{2}$ we would have a $k$-dimensional singularity with Hausdorff $k$-dimensional measure $0$.