Sharp quantitative stability of the Ashbaugh--Benguria inequality

Abstract

We establish sharp quantitative stability of the Ashbaugh--Benguria inequality on bounded convex domains in ℝⁿ. After volume and centroid normalization, the eigenvalue-ratio deficit controls the squared H^−1/2(𝕊ⁿ⁻¹) norm of the radial displacement from the unit sphere. This yields the optimal exponent 5/2 for the Fraenkel asymmetry and, for small deficits, the optimal exponent (n+4)/2 for the normalized Hausdorff distance. All three exponents are sharp even among smooth strictly convex domains whose principal curvatures lie in [1/2,2]. In particular, the quadratic Fraenkel-asymmetry estimate proposed by Brasco and De Philippis (2017, Open Problem 7.65) fails. We also determine the optimal stability exponents under higher-order radial regularity bounds.

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BibTeX

@misc{huang2026sharp,
  title = {{Sharp quantitative stability of the Ashbaugh--Benguria inequality}},
  author = {Long-Han Huang and Yuxuan Zhou and Wenming Zou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31223},
  eprint = {2610.31223},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31223},
}

RIS

TY  - UNPB
TI  - Sharp quantitative stability of the Ashbaugh--Benguria inequality
AU  - Long-Han Huang
AU  - Yuxuan Zhou
AU  - Wenming Zou
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31223
UR  - https://arxiv.org/abs/2610.31223
AB  - We establish sharp quantitative stability of the Ashbaugh--Benguria inequality on bounded convex domains in ℝ^n. After volume and centroid normalization, the eigenvalue-ratio deficit controls the squared H^−1/2(𝕊^n−1) norm of the radial displacement from the unit sphere. This yields the optimal exponent 5/2 for the Fraenkel asymmetry and, for small deficits, the optimal exponent (n+4)/2 for the normalized Hausdorff distance. All three exponents are sharp even among smooth strictly convex domains whose principal curvatures lie in [1/2,2]. In particular, the quadratic Fraenkel-asymmetry estimate proposed by Brasco and De Philippis (2017, Open Problem 7.65) fails. We also determine the optimal stability exponents under higher-order radial regularity bounds.
ER  -

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