On a Bonnesen type inequality involving the spherical deviation
On a Bonnesen type inequality involving the spherical deviation
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DOI：
10.1016/j.matpur.2012.05.006
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发表时间：
2012-12
期刊：
影响因子：
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通讯作者：
N. Fusco;M. Gelli;Giovanni Pisante
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文献类型：
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作者：
N. Fusco;M. Gelli;Giovanni Pisante
本文研究了满足温和正则性的有限周长集的球面偏差相对于等周亏量的稳定性，推广了Fuglede的经典Bonnesen型结果在近球域上的非凸集.特别地，我们证明了如果一组有限周长E满足一个具有足够宽角的内锥条件(参看.定义2)然后我们有λH(E)⩽Φ(D(E))，其中λH(E)是相对于Hausdorff距离的球面形状的偏差，D(E)表示等周亏量，Φ是依赖于维度在零点连续消失的显式函数。
In this paper we investigate the stability of the deviation from being a sphere with respect to the isoperimetric deficit for sets of finite perimeter satisfying a mild regularity property, giving an extension to non-convex sets of the classical Bonnesen type result of Fuglede for nearly spherical domains. In particular we prove that if a set of finite perimeter E satisfies an interior cone condition with sufficiently wide angles (cf. Definition 2.1) then we have λ H (E)⩽ Φ (D (E)), where λ H (E) is the deviation from a spherical shape with respect to the Hausdorff distance, D (E) denotes the isoperimetric deficit and Φ is an explicit function vanishing continuously at zero and depending on the dimension.