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 Rⁿ⁺¹, {align*} ẋ=({1}{{E_k(κ̂)}{Eₖ₋₁(κ̂)}-α }- x,ν)ν, k=2,3,…,n-1. {align*} Assuming that the initial hypersurface M₀ ⊂ Rⁿ⁺¹ is star-shaped and its shifted principal curvatures κ̂=κ+α(1,…,1) lie in the convex set {align*} Γα,k:=Γₖ₋₁∩ \{λ∈ R^n:\, E_k(λ)-α Eₖ₋₁(λ)>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.