Relating diameter and mean curvature for submanifolds of Euclidean space (Q936132)

The author proves the following theorem which is their main result: For \(n>m\geq1\), suppose that \(\mathcal{M}\) is an \(m\)-dimensional closed (compact, no boundary) connected manifold smoothly immersed \(\mathbb{R}^{n}\). Then its intrinsic diameter \(d_{\text{int}}\) and its mean curvature \(\mathbf{H}\) are related by \[ d_{\text{int}}\leq C(m)\int_{\mathcal{M}}| \mathbf{H}| ^{m-1}d\mu \] where the intrinsic diameter \(d_{\text{int}}\) is defined by \(d_{\text{int}}:=\text{max}_{x,y\in \mathcal{M}}\text{dist}_{\mathcal{M}}(x,y)\).