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.


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.


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