The quermassintegral inequalities for k-convex spiral-star-shaped domains

Abstract

We extend the quermassintegral inequalities of our earlier work \cite{GuanLi2009} from star-shaped domains to a substantially broader class of domains in Euclidean space, which we call spiral-star-shaped. A smooth domain is spiral-star-shaped if its boundary satisfies ⟨ X+A X,ν⟩>0 for some fixed skew-symmetric linear map $ A$, where X is the position vector and ν is the outward unit normal. The vector field X+A X is conformal Killing, with dilation component X and rotational component $A X$; its trajectories combine expansion and rotation, giving rise to the spiral terminology. When A=0, it reduces to ordinary star-shapedness, while nonzero A allows domains that are not star-shaped in the usual radial sense. For smooth k-convex spiral-star-shaped domains, we prove long-time existence and exponential convergence to a round sphere for a locally constrained inverse curvature flow. The positive spiral support function provides the geometric control needed to treat initial hypersurfaces that need not be star-shaped in the usual sense. Monotonicity of the quermassintegrals along the flow yields sharp quermassintegral inequalities in this enlarged class, with equality if and only if the domain is a ball.

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BibTeX RIS

BibTeX

@misc{li2026quermassintegral,
  title = {{The quermassintegral inequalities for k-convex spiral-star-shaped domains}},
  author = {Junfang Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35394},
  eprint = {2610.35394},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35394},
}

RIS

TY  - UNPB
TI  - The quermassintegral inequalities for k-convex spiral-star-shaped domains
AU  - Junfang Li
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35394
UR  - https://arxiv.org/abs/2610.35394
AB  - We extend the quermassintegral inequalities of our earlier work \cite{GuanLi2009} from star-shaped domains to a substantially broader class of domains in Euclidean space, which we call spiral-star-shaped. A smooth domain is spiral-star-shaped if its boundary satisfies ⟨ X+A X,ν⟩>0 for some fixed skew-symmetric linear map $ A$, where X is the position vector and ν is the outward unit normal. The vector field X+A X is conformal Killing, with dilation component X and rotational component $A X$; its trajectories combine expansion and rotation, giving rise to the spiral terminology. When A=0, it reduces to ordinary star-shapedness, while nonzero A allows domains that are not star-shaped in the usual radial sense. For smooth k-convex spiral-star-shaped domains, we prove long-time existence and exponential convergence to a round sphere for a locally constrained inverse curvature flow. The positive spiral support function provides the geometric control needed to treat initial hypersurfaces that need not be star-shaped in the usual sense. Monotonicity of the quermassintegrals along the flow yields sharp quermassintegral inequalities in this enlarged class, with equality if and only if the domain is a ball.
ER  -

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