arXiv subjects
Publications and source records attributed to Xiong Ge.
For solutions to the Minkowski problem of the even dual curvature measures $\widetilde{C}_q$ in $\mathbb{R}^n$, $n\ge 2$, we prove the non-uniqueness for $q>n$ and $n\ge 2$, and the uniqueness for $0<q<n$ and $n=2$. These results are governed by our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals $\widetilde V_q$.
In the 1960s, Gr\"unbaum and Loewner raised problems concerning the minimum possible number of hyperplane sections of a convex body in $R^n$ whose centroids coincide with the centroid of the body itself. By employing the hemispherical transform and Morse theory, we provide a complete solution to these problems.
In this paper, we give a description of the John contact points of a regular simplex. We prove that the John ellipsoid of any simplex is ball if and only if this simplex is regular and that the John ellipsoid of a regular simplex is its inscribed ball.