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<p>Stability results reflect the degree of deviation between geometric objects and the solutions to uniqueness problems under certain metrics in convex geometry. In this paper, we explore a new type of stability: given a measure <i>μ</i> on ℝ<sup><i>n</i>−1</sup> with a positive continuous density, we aim to identify the conditions under which one can estimate the distance (e.g., Hausdorff distance or radial distance) between two origin-symmetric star bodies based on the distance between their respective <i>μ</i>-measure-dependent intersection bodies. More precisely, we establish corresponding stability estimates for this problem using spherical harmonic expansions and the Poisson transform.</p>

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Stability of intersection bodies related to measures with densities

  • 期刊论文

  • 发表日期：

    2026年3月5日

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  • Tian Li,
  • Baocheng Zhu

摘要

Stability results reflect the degree of deviation between geometric objects and the solutions to uniqueness problems under certain metrics in convex geometry. In this paper, we explore a new type of stability: given a measure μ on ℝn−1 with a positive continuous density, we aim to identify the conditions under which one can estimate the distance (e.g., Hausdorff distance or radial distance) between two origin-symmetric star bodies based on the distance between their respective μ-measure-dependent intersection bodies. More precisely, we establish corresponding stability estimates for this problem using spherical harmonic expansions and the Poisson transform.