New quermassintegral and Poincaré type inequalities for non-convex domains

Abstract

In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space ${\mathbb{R}}^{n+1}$, \begin{align*} \dot{x}=\left(\frac{1}{\frac{E_k(\hatκ)}{E_{k-1}(\hatκ)}-α}-\langle x,ν\rangle\right)ν, \quad k=2,3,\ldots,n-1. \end{align*} Assuming that the initial hypersurface ${\mathcal{M}}_0\subset {\mathbb{R}}^{n+1}$ is star-shaped and its shifted principal curvatures $\hat{\kappa }=\kappa +\alpha (1,\dots ,1)$ lie in the convex set \begin{align*} Γ_{α,k}:=Γ_{k-1}\cap \{λ\in \mathbb{R}^n:\, E_k(λ)-αE_{k-1}(λ)>0\}, \end{align*} we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains. In the second part, we derive a Poincaré type inequality for $k$-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.