# Volume and Gromov-Hausdorff limits of hypersurfaces in $\mathbb{R}^{n+1}$

I have 3 questions:

1) Let $$(M_j^n,p_j)_{j\in \mathbb{N}}$$ be a sequence of pointed mean-convex embedded hypersurfaces in $$\mathbb{R}^{n+1}$$. Does a subsequence converge in the Gromov-Hausdorff topology to some pointed metric space $$(X,d,p)$$?

2) More generally, given a constant $$k\in \mathbb{R}$$ and a sequence $$(M_j^n,p_j)_{j\in \mathbb{N}}$$ of pointed embedded hypersurfaces in $$\mathbb{R}^{n+1}$$ with mean curvature $$\geq k$$, does a subsequence converge in the Gromov-Hausdorff topology to some pointed metric space $$(X,d,p)$$?

According to Gromov's precompactness theorem, 1) and 2) are true iff there exists a function $$\mathcal{N}:(0,\infty)^2\to (0,\infty)$$ such that, for every $$j,\epsilon,r>0$$, the maximum number of disjoint balls of radius $$\epsilon$$ contained inside of $$B_r(p_j)\subset M_j^n$$ is bounded above by $$\mathcal{N}(\epsilon,r)$$. (See for example Petersen's book Riemannian Geometry.)

However, I am not sure whether such a function exists.

The analogous statement for Ricci curvature bounded below is proven using Bishop-Gromov volume monotonicity, which leads to the last question:

3) Is there an analogue of volume monotonicity for hypersurfaces with mean curvature bounded below? My guess is that the comparison would be against constant curvature hypersurfaces like the round sphere.