The Mahler conjecture in dimension three

发布时间：2026-06-10 作者：浏览次数：

Speaker:	李媛媛	DateTime:	2026年6月11日（周四）下午15:00-15:40
Brief Introduction to Speaker: 李媛媛，西湖大学		Place:	国交2号楼315会议室
Abstract:We settle the three-dimensional Mahler conjecture for arbitrary convex bodies. Prior to this work, the Mahler conjecture for arbitrary convex bodies had remained open in all dimensions n≥3 since Mahler's original work in 1938. More precisely, for every convex body K ⊆ R^3, we prove the sharp inequality \|K\|\|K^{s(K)}\|≥649 , where K^{s(K)} denotes the polar of K with respect to Santal'o point s(K). The lower bound is attained by simplices; among polytopes, these are the only equality cases. The key ingredient in our proof is an admissible shadow-system framework, which produces face-lattice-preserving, volume-affine deformations and reduces the problem to a dimension count for the space of admissible speeds. As an application, this framework also yields a purely geometric new proof of the three-dimensional centrally symmetric Mahler inequality, first proved by Iriyeh and Shibata. This is joint work with Shibing Chen (USTC)，Dongmeng Xi(SHU) and Zhe-Feng Xu(SISSA&USTC).