Sharp geometric inequalities for the general $p$-affine capacity

Han Hong Memorial University of Newfoundland Deping Ye Memorial University of Newfoundland

Metric Geometry Convex and Discrete Geometry mathscidoc:1904.23001

J. Geom. Anal., 28, (3), 2254-2287, 2018.7
In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-variational capacity and the $p$-integral affine surface area. Consequently, several sharp geometric inequalities for the general $p$-affine capacity are obtained. These inequalities extend and strengthen many well-known (affine) isoperimetric and (affine) isocapacitary inequalities.
Asymmetric $L_p$ affine Sobolev inequality, general $L_p$ affine isoperimetric inequality, isocapacitary inequality, $L_p$ affine isoperimetric inequality, $L_p$ affine Sobolev inequality, $L_p$ projection body, $p$-affine capacity, $p$-integral affine surface area, $p$-variational capacity
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  title={Sharp geometric inequalities for the general $p$-affine capacity},
  author={Han Hong, and Deping Ye},
  booktitle={J. Geom. Anal.},
Han Hong, and Deping Ye. Sharp geometric inequalities for the general $p$-affine capacity. 2018. Vol. 28. In J. Geom. Anal.. pp.2254-2287.
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