# convergence and a mean curvature condition imply convexity

I have a question regarding the proof of theorem 4.6 in https://arxiv.org/abs/1007.3899 (Hall's conjecture).

Let $$S^2$$ be the class of Borel subsets in $$\mathbb{R}^2$$ with finite and positive Lebesgue measure $$|\, .\, |$$. For a given set $$E\in S^2$$ we denote with $$B_E$$ the ball centered in the origin with $$|E|=|B_E|$$, and $$P(E)$$ the perimeter of $$E$$ (see https://en.wikipedia.org/wiki/Caccioppoli_set).

For $$E\in S^2$$ let $$\delta P(E):=\frac{P(E)-P(B_E)}{P(B_E)}$$ and $$\alpha(E):=\inf\limits_{x\in\mathbb{R}^n}\frac{|E\triangle (x+B_E)|}{|B_E|}$$. And for $$E\in S^2$$ with $$\alpha(E)>0$$ define $$Q(E):=\frac{\delta P(E)}{\alpha(E)^2}$$.

In the proof of theorem 4.6 it is claimed: If $$(E_j)_j\subset S^2$$ is a sequence of sets such that $$\lim\limits_{j\to\infty}Q(E_j)= Q(B) \text{ and } \lim\limits_{j\to\infty}\|H_j-1\|_{L^\infty(\partial E_j)}=0,$$ where $$H_j$$ stands for the mean curvature of the boundary $$\partial E_j$$ and $$B:=B_1(0)$$, then there exists a $$j_0>0$$ such that $$E_j$$ is a convex set for all $$j\ge j_0$$.

Does anyone know why that holds (i.e. how the convexity is concluded)? Or do you know any reference for an anything which helps to show the convexity? I appreciate any help.

Edit: I have found the book "Lecture Notes on Mean Curvature Flow " by Carlo Mantegazza, where in the section 4.5.1 about "an alternative proof of Grayson's theorem" there can be found a similar argument (in the last section before remark 4.5.8), namely roughly: Let $$\gamma_t$$ be the curvature flow of a closed, embedded, smooth curve in the plane, $$\tilde{\gamma}_{q_i}$$ a certain new sequence of rescaled curves with uniformly bounded curvatures in $$L^2$$, such that a certain subsequence converges in $$C^2$$ to a limit curve which is a unit circle. Assuming that the $$\tilde{\gamma}_{q_i}$$ has positive curvature, then they curves are convex.

Thus, I think there is a general fact which I don't know (I am not an expert in this field), giving convexity under assumptions that some sequence converges to the unit ball and an extra assumption abouts it's curverture.

## 1 Answer

Actually one does not use the convergence of $$Q(E_j)$$, but one implicitly use (iii) of theorem 3.2: your sets are the interiors of Jordan curves that are $$C^1$$ close to the unit circle. Then positive curvature implies convexity, and since the curvature goes to $$1$$ uniformly, it is eventually positive.

To see that positive curvature of a Jordan curve implies convexity, look at any $$x\in E_j$$ and consider the subset $$E_j'$$ of points $$y$$ such that the geodesic segment $$[xy]$$ is in $$E_j$$. If this is not the whole of $$E_j$$, then you can find an interval $$(xy)$$ that touches the boundary tangentially. This is in contradiction with the positivity of curvature.