I have been thinking about the following question and have been unable to find any literature on the subject.

Question: Assume I have a sequence of smooth, simply connected, compact domains $\Omega_s \subset \mathbb{R}^d$ such that $|\Omega_s|=1$ and

$\int_{\partial \Omega_s} (\kappa - \bar \kappa)^2 dS(y) \to 0$ as $s \to +\infty$,

where here $\kappa$ is the mean curvature of the surface $\partial \Omega_s$ and $\bar \kappa$ denotes the average mean curvature over $\partial \Omega_s$. I can prove that the limit is in fact a ball in the following two cases:

1. All of the sets $\Omega_s$ are convex. or
2. I assume the uniform bound $\limsup_{s \to +\infty} |\partial \Omega_s| + \int_{\partial \Omega_s} \kappa^2 dS < +\infty$.

I would however like to remove these restrictions since they seem quite artificial. I have been able to rule out the standard "pinching" counter examples of a long rod with capped ends, but am not sure if there could exist other pathologies. Any direction to results in this direction would be appreciated.

I have been thinking about the following question and have been unable to find any literature on the subject.

Question: Assume I have a sequence of smooth, simply connected, compact domains $\Omega_s \subset \mathbb{R}^d$ such that

$\int_{\partial \Omega_s} (\kappa - \bar \kappa)^2 dS(y) \to 0$ as $s \to +\infty$,

where here $\kappa$ is the mean curvature of the surface $\partial \Omega_s$ and $\bar \kappa$ denotes the average mean curvature over $\partial \Omega_s$. I can prove that the limit is in fact a ball in the following two cases:

1. All of the sets $\Omega_s$ are convex. or
2. I assume the uniform bound $\limsup_{s \to +\infty} |\partial \Omega_s| + \int_{\partial \Omega_s} \kappa^2 dS < +\infty$.

I would however like to remove these restrictions since they seem quite artificial. I have been able to rule out the standard "pinching" counter examples of a long rod with capped ends, but am not sure if there could exist other pathologies. Any direction to results in this direction would be appreciated.