Assume by translation that $0$ is the center of mass of $S_{h,p}(x)$. By subtracting a linear function we can assume that

$$p=0,\quad u|_{\partial S_{h,0}(x)}\leq 0,\text{ and }\quad|\min_{S_{h,0}(x)}u|=h.$$

By John’s Lemma, there is a linear transformation $A$ that normalizes $S_{h,0}(x)$. Let

$$\tilde{u}(x)=|\det A|^{-2/n}u(Ax).$$

It is easy to check that

$$\det D^{2}\tilde{u}\geq 1,\quad\tilde{u}|_{\partial\tilde{\Omega}}\leq 0$$

where $B_{1}\subset\tilde{\Omega}\subset B_{C(n)}$. Then $\frac{1}{2}(|x|^{2}-1)$ is an upper barrier for $\tilde{u}$, so

$$|\min_{\tilde{\Omega}}\tilde{u}|\geq\frac{1}{2}.$$

Since $|\det A|\geq c(n)|S_{h,0}(x)|$, the conclusion follows. ∎