Uniqueness And Ellipsoid Solutions Of Christoffel-Minkowski Problem Of Firey’s ρ-sum

In this paper, we study the following Christoffel-Minkowski problem of Firey’s p-sum where H is the support function of convex bodies, Sn is the unit sphere of n dimension,f is a positive function defined on Sn, k ∈{1,2, …, n} and p ∈R. We obtain that the uniqueness of (Ⅰ) holds when p∈R+\{k}. Furthermore, the solution of (Ⅰ) is unique upto a dilatation when p=k, i.e., if H solves (I),{aH:a ∈R+} are the whole solutions of (Ⅰ).Then we substitute the binomial coefficient Cnk for f in (Ⅰ). Starting from the geometric perspective, we study the ellipsoid solutions of through the support function of convex bodies. We obtain that, when the solution B of (Ⅱ) is an ellipsoid, there are only 3 cases:Case 1:when p=k, B is an arbitrary ball;Case 2:when(p,k)= (-n-2, n), the product of all the half-axis of the ellipsoid is 1, which shows that the volume of B is a constant ωn+i, where ωn+1 is the volume of the unit ball in Rn+1;Case 3:in the rest cases, B is certainly a unit ball.