Abstract
Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p≥1 in ℝ2 in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in ℝ2 such that it has area π and its p-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, K is close to an ellipse in the Hausdorff distance.
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Acknowledgements
I would like to thank Alina Stancu and Károly Böröczky for comments and suggestions that have improved the initial manuscript. I am indebted to two referees for the very careful reading of the original submission.
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Communicated by Ben Andrews.
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Ivaki, M.N. On the Stability of the p-Affine Isoperimetric Inequality. J Geom Anal 24, 1898–1911 (2014). https://doi.org/10.1007/s12220-013-9401-1
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DOI: https://doi.org/10.1007/s12220-013-9401-1
Keywords
- Affine support function
- Affine normal flow
- Hausdorff distance
- Stability of geometric inequalities
- p-affine surface area
- p-affine isoperimetric inequality