Geodesic
An estimate from below for the spatial diameter of a~surface in terms of its intrinsic radius and curvature
Sbornik. Mathematics, Tome 15 (1971) no. 3, pp. 405-414

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In this paper we prove the following Theorem. Let $F$ be a regular simply connected surface of class $C^3$ in $R^3$. There exist postitive absolute constants $C$ and $C_1$ such that if $$ \mu=\int_F|K|\,dS, $$ where $K$ is the Gaussian curvature and $S$ is the area element on $F$, the estimate $$ d\geqslant\bigl(\sqrt3-C_1\sqrt\mu\bigr)r $$ holds. Bibliography: 11 titles. 
@article{SM_1971_15_3_a2,
     author = {Yu. D. Burago},
     title = {An estimate from below for the spatial diameter of a~surface in terms of its intrinsic radius and curvature},
     journal = {Sbornik. Mathematics},
     pages = {405--414},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {1971},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_15_3_a2/}
}
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Yu. D. Burago. An estimate from below for the spatial diameter of a~surface in terms of its intrinsic radius and curvature. Sbornik. Mathematics, Tome 15 (1971) no. 3, pp. 405-414. http://geodesic.mathdoc.fr/item/SM_1971_15_3_a2/