On the stability of the $L_p$-curvature

It is known that the $L_p$-curvature of a smooth, strictly convex body in $\mathbb{R}^{n}$ is constant only for origin-centred balls when $1\neq p>-n$, and only for balls when $p=1$. If $p=-n$, then the $L_{-n}$-curvature is constant only for origin-symmetric ellipsoids. We prove `local' and `global' stability versions of these results. For $p\geq 1$, we prove a global stability result: if the $L_p$-curvature is almost a constant, then the volume symmetric difference of $\tilde{K}$ and a translate of the unit ball $B$ is almost zero. Here $\tilde{K}$ is the dilation of $K$ with the same volume as the unit ball. For $0\leq p<1$, we prove a similar result in the class of origin-symmetric bodies in the $L^2$-distance. In addition, for $-n<p<0$, we prove a local stability result: There is a neighborhood of the unit ball that any smooth, strictly convex body in this neighborhood with `almost' constant $L_p$-curvature is `almost' the unit ball. For $p=-n$, we prove a global stability result in $\mathbb{R}^2$ and a local stability result for $n>2$ in the Banach-Mazur distance.