New quermassintegral and Poincar\'e type inequalities for non-convex domains

In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space R^n+1, align* ẋ= (1Eₖ ({) E₊-₁ () - }- x, ), k=2, 3, , n-1. align* Assuming that the initial hypersurface M₀ R^n+1 is star-shaped and its shifted principal curvatures =+ (1, , 1) lie in the convex set align*, ₊: =₊-₁ \ Rⁿ: \, Eₖ () - E₊-₁ () >0\, 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 Poincar\'e type inequality for k-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.