In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.
Aleksandrov, Existence and uniqueness of a convex surface with a given integral curvature, C
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The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem
In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.