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 \[ \langle X+A X,ν\rangle>0 \] 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.