An estimate from below for the spatial diameter of a surface in terms of its intrinsic radius and curvature

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.
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