The dual Minkowski problem under group actions

Abstract

In this paper, we study the dual Minkowski problem under group symmetry. For 0 < 𝑞 ≤ 𝑛 , we give a complete existence characterization in the framework of G -invariant convex bodies when the group 𝐺 ⊂ 𝑂 ⁡ ( 𝑛 ) has no nonzero fixed points, recovering the origin-symmetric setting when 𝐺 = { ± 𝐼 } . The necessary and sufficient conditions concern the concentration of the measure on G -invariant subspaces, both in the range 0 < 𝑞 < 𝑛 and at the critical endpoint 𝑞 = 𝑛 , where the problem becomes the logarithmic Minkowski problem.