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In this paper we study the solvability of the rotationally symmetric centroaffine Minkowski problem. By delicate blow-up analyses, we remove a technical condition in the existence result obtained by Lu and Wang [30].

We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature K_N with speed given by F^{−α}, where α∈(0,1] for K_N=0,−1 and α=1 for K_N=1, F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual F_∗ approaching zero on the boundary of the positive cone Γ_+. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to 1. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.

In this paper we show by example that there is no uniform estimate for the L p-Minkowski problem. As a result we obtain the nonuniqueness of solutions to the problem.