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Intrinsic diameter and curvature integrals of surfaces immersed in Rn

We prove that if Σ is a complete connected surface without boundary smoothly immersed in R n , then its intrinsic diameter satisfies the bound diam Σ < C n vol N(Σ), where N(Σ) ⊂ R n x S n-1 is the manifold of unit normals to Σ and C n is a universal constant depending only on n. In fact we establish a stronger bound, linear in the sum of the area of Σ and the L 1 norms of its second fundamental form and Gauss curvature (in dimension n = 3 the two bounds are equivalent).