Paper · Oct 26, 2026
The quermassintegral inequalities for k-convex spiral-star-shaped domains
As on arXiv
Abstract
We extend the quermassintegral inequalities of our earlier work 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 for some fixed skew-symmetric linear map , where is the position vector and is the outward unit normal. The vector field is conformal Killing, with dilation component and rotational component ; its trajectories combine expansion and rotation, giving rise to the spiral terminology. When , it reduces to ordinary star-shapedness, while nonzero allows domains that are not star-shaped in the usual radial sense. For smooth -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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