For every nonzero Borel measure not concentrated in a closed hemisphere, there is a convex body whose Lq anisotropic p-torsional measure equals the measure, up to a constant when 0<q<1.
Chen, C Li and W Wang, The Lp Gauss dual Minkowski problem , arXiv:2412.13557, 2024
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The Lq-Minkowski problem of anisotropic p-torsional rigidity
For every nonzero Borel measure not concentrated in a closed hemisphere, there is a convex body whose Lq anisotropic p-torsional measure equals the measure, up to a constant when 0<q<1.