The Journal of Geometric Analysis

, Volume 24, Issue 4, pp 1898–1911

# On the Stability of the p-Affine Isoperimetric Inequality

• Mohammad N. Ivaki
Article

## 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.

## Keywords

Affine support function Affine normal flow Hausdorff distance Stability of geometric inequalities p-affine surface area p-affine isoperimetric inequality

## Mathematics Subject Classification (2010)

52A40 53C44 53A04 52A10 53A15 53A15

## Notes

### 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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