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 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $$ \mu=\int_F|K|\,dS<C, $$ 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.

[1] A. D. Aleksandrov, V. A. Zalgaller, Dvumernye mnogoobraziya ogranichennoi krivizny, Trudy Matem. in-ta im. V. A. Steklova, 63, 1962 | MR | Zbl

[2] R. Bellman, Teoriya ustoichivosti reshenii differentsialnykh uravnenii, IL, Moskva, 1954

[3] Yu. D. Burago, M. B. Stratilatova, “Okruzhnost na poverkhnosti”, Trudy Matem. in-ta im. V. A. Steklova, 76, 1965, 88–114 | MR

[4] Yu. D. Burago, Neravenstva izoperimetricheskogo tipa v teorii poverkhnostei ogranichennoi vneshnei krivizny, Zapiski nauchnykh seminarov LOMI, 10, 1968 | MR | Zbl

[5] Yu. D. Burago, S. Z. Shefel, “Neregulyarnye poverkhnosti ogranichennoi vneshnei krivizny i neravenstva izoperimetricheskogo tipa”, DAN SSSR, 182:5 (1968), 997–999 | MR | Zbl

[6] S. S. Chern, R. K. Lashof, “On the total curvature of immersed manifolds”, Amer. J. Math., 79:2 (1957), 306–313 | DOI | MR

[7] I. Fary, “Sur certaines inegalites geometriquis”, Acta Scient. Math., 12 (1950), 117–124 | MR | Zbl

[8] L. A. Santalo, “On some geometric inequalities in the style of Fary”, Amer. J. Math., 41:1 (1969), 25–31 | DOI | MR

[9] H. Whitney, “On singularities of mapings of euclidean spaces”, Ann. Math., 62:3 (1955), 374–410 | DOI | MR | Zbl

[10] F. J. Almgren, “An isoperimetric inequality”, Proc. Amer. Math. Soc., 15:2 (1964), 284–285 | DOI | MR | Zbl

[11] S. Eilenberg, O. G. Harrold, “Continua of finite linear measure 1”, Amer. J. Math., 65:1 (1943), 137–146 | DOI | MR | Zbl