We study the motion of smooth, closed, strictly convex hypersurfaces in R n+1 expanding in the direction of their normal vector field with speed depending on the kth elementary symmetric polynomial of the principal radii of curvature σ k and support function h. A homothetic self-similar solution to the flow that we will consider in this paper, if exists, is a solution of the well-known Lp-Christoffel-Minkowski problem ϕh 1−p σ k = c. Here ϕ is a preassigned positive smooth function defined on the unit sphere, and c is a positive constant. For 1 ≤ k ≤ n − 1, p ≥ k + 1, assuming the spherical hessian of ϕ where p ∈ R, E i is the ith elementary symmetric polynomial of principal curvatures for 0 ≤ i ≤ n normalized so that E i (1, . . . , 1) = 1 (and E 0 ≡ 1), ν(·, t) is the outer unit normal vector of M t := F (M n , t) and ϕ is a positive smooth function defined on the unit sphere S n .Assuming M t is strictly convex, its support function as a function on the unit sphere is given byWriteḡ and∇ for the standard round metric and the Levi-Civita connection of S n . Recall that the principal radii of curvature are the eigenvalues of the matrix r ij :=∇ i∇j h +ḡ ij h

We study long-time existence and asymptotic behavior for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even Lp-Minkowski problems in R n+1 for p > −n − 1.

We study the motion of smooth, strictly convex bodies in R n expanding in the direction of their normal vector field with speed depending on Gauss curvature and support function.

Abstract. We study the long time behavior of the volume preserving p-flow in R n+1 for 1 ≤ p < n+1 n−1. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving p-flow converges sequentially to the unit ball in the C ∞ topology, modulo the group of special linear transformations.

Abstract. We consider two types of p-centro affine flows on smooth, centrally symmetric, closed convex planar curves, p-contracting, respectively, pexpanding. Here p is an arbitrary real number greater than 1. We show that, under any p-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area π converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a p-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any p-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized as to enclose constant area π, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to SL(2) transformations, to the unit circle. At the end, we present a new proof of p-affine isoperimetric inequality, p ≥ 1, for smooth, centrally symmetric convex bodies in R 2 .