Relating diameter and mean curvature for submanifolds of Euclidean space

Abstract

Given a closed $m$ -dimensional manifold $M$ immersed in $R^{n}$ , we estimate its diameter $d$ in terms of its mean curvature $H$ by

d \leq C (m) \int_{M} ∣ H ∣^{m - 1} d μ .

Peter Topping, Relating diameter and mean curvature for submanifolds of Euclidean space. Comment. Math. Helv. 83 (2008), no. 3, pp. 539–546