Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space Rⁿ+¹ with speed f r^αK, where K is the Gauss curvature, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. If α ≥ n + 1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if f ≡ 1. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang [30] for q < 0. If α < n + 1, corresponding to the case q > 0, we also establish the same results for even functions f and origin-symmetric initial conditions, but for f non-symmetric, a counterexample is given to the above smooth convergence.