论文标题
关于$ l_p $ - 续签的稳定性
On the stability of the $L_p$-curvature
论文作者
论文摘要
众所周知，在$ \ mathbb {r}^{n} $中，$ l_p $ c的光滑，严格凸出的主体仅在$ 1 \ neq p> -n $时才适用于以原点为中心的球，并且仅在$ p = 1 $时球球。如果$ p = -n $，则$ l _ { - n} $ - 曲率仅适用于原点对称椭圆形。我们证明了这些结果的“本地”和“全球”稳定版本。对于$ p \ geq 1 $，我们证明了一个全球稳定性结果：如果$ l_p $ - 纯相几乎是一个常数，则$ \ tilde {k} $的体积对称差，并且单位球$ b $的翻译几乎为零。这里$ \ tilde {k} $是$ k $的扩张，其体积与单位球相同。对于$ 0 \ leq p <1 $，我们证明了$ l^2 $ distance的原点对称物体的类似结果。此外，对于$ -n <p <0 $，我们证明了局部稳定性结果：单位球附近，在这个社区中，任何光滑，严格地凸出的身体都具有“几乎”常数$ l_p $ curvature，是单位球的“几乎”。对于$ p = -n $，我们证明了$ \ mathbb {r}^2 $的全局稳定性结果，而在Banach-Mazur距离中，$ n> 2 $的本地稳定性结果。
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.
