Limiting shape of the $L_p$-Minkowski problem

Ben Andrews classified the limiting shape for isotropic curvature flow corresponding to the solutions of the $L_p$-Minkowski problem as $p\to -\mathrm{\infty }$ in the planar case. In this paper, we use the group-invariant method to study the asymptotic shape of solutions to the $L_p$-Minkowski problem as $p\to -\mathrm{\infty }$ in high dimensions. For any regular polytope $T$, we establish the existence of a solution ${\mathrm{\Omega }}^{(p)}$ to the $L_p$-Minkowski problem that converges to $T$ as $p\to -\mathrm{\infty }$, thereby revealing the intricate geometric structure underlying this limiting behavior. We also extend the result to the dual Minkowski problem.