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In this Paper there is a proof that a closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius $L/4 - (\pi - 2)/2K$. Are there similar results for smooth surfaces in $\mathbb{R}^3$?

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    $\begingroup$ The bound can become negative for small values of $K$ and fixed $L$? $\endgroup$ – Loïc Teyssier May 15 '16 at 22:36
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    $\begingroup$ From the Chakerian et al. paper: In 2D, the extreme is a racetrack, the convex hull of two circles of the same "radius $1/K$ whose centers are a distance $L/2 - \pi/K$ apart." $\endgroup$ – Joseph O'Rourke May 15 '16 at 23:27
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    $\begingroup$ @LoïcTeyssier A negative bound here is not a priori absurd. If $K$ is too small compared to $L$, there may not be any closed curves of the sort in the question --- the ends of a segment of length $L$ can't be brought together without introducing a non-negligible amount of curvature. $\endgroup$ – Andreas Blass May 15 '16 at 23:42
  • $\begingroup$ My bad, I didn't read the key word "closed". Now it makes sense. $\endgroup$ – Loïc Teyssier May 16 '16 at 7:16
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A possible generalization is an upper bound for the diameter of a surface $S \subset \mathbb{R}^3$ in terms of its area $A(S)$ and its gaussian and mean curvatures $K$, $H$. Let me give an answer in the case of positively curved surfaces.

We define an ovaloid as a compact connected surface $S \subset \mathbb{R}^3$ with $K(p) >0$ for all $p \in S$. If $S$ is an ovaloid then it is orientable, and by using Hadamard-Stoker theorem it follows that its inner domain is a convex subset of $\mathbb{R}^3$ and that its Gauss map $N \colon S \to \mathbb{S}^2$ is a diffeomorphism. In particular, $S$ is a topological sphere.

The following classical estimate for the diameter of an ovaloid is due to Bonnet. Modern proofs are based on Hopf-Rinow theorem, see for instance

S. Montel, A. Ros, Curves and Surfaces, Theorem 7.45.

Theorem (Bonnet). Let $S \subset \mathbb{R}^3$ be a closed connected surface whose Gaussian curvature $K$ satisfies $\inf_{p \in S} K(p) >0$. Then $S$ is compact, that is $S$ is an ovaloid, and its diameter satisfies $$\textrm{diam} \, S \leq \frac{\pi}{\sqrt{\inf_{p \in S} K(p)} }.$$

Now, by using the isoperimetric inequality we can show that for any ovaloid the inequality $$\bigg(\int_S H(p) \bigg)^2 \geq 4 \pi A(S)$$ holds, see Exercise 10 p. 196 in the book quoted above. Substituting into the inequality provided by Bonnet's theorem, we finally obtain $$\textrm{diam} \, S \leq \frac{\big(\int_S H(p) \big)^2}{4 \, A(S) \sqrt{\inf_{p \in S} K(p)} }.$$

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Sufficient conditions, in terms of geometric invariants of the domains, such as volumes, surface areas and curvature integrals of the boundaries of domains, under which one domain can be moved by an isometry inside the other domain in a 3-dimensional space of constant curvature, are given in http://www.jstor.org/stable/119206 (Sufficient Conditions for One Domain to Contain Another in a Space of Constant Curvature, by J. Zhou).

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