发表机构
Università di Padova; Università di Pisa（帕多瓦大学; 比萨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了三维平行多面体中，正则截角八面体是等体积下表面积最小的唯一平行多面体，非截角的Fedorov类型中尖锐界由正则菱形十二面体唯一达到。
AI 中文摘要
我们证明了，在所有固定体积的平行多面体中，正则截角八面体的表面积最小且是唯一的。等价地，每个三维平行多面体P都满足∂P的二维豪斯多夫测度除以|P|的三分之二次方大于等于3(1+2√3)除以4的三分之二次方，当且仅当P与正则截角八面体相似时等号成立。在非截角的Fedorov类型中，我们证明了一个更强的尖锐界，该界由正则菱形十二面体唯一达到。
英文摘要
We prove that the regular truncated octahedron uniquely minimizes surface area among all parallelohedra of fixed volume. Equivalently, every three-dimensional parallelohedron $P$ satisfies \[ \frac{\mathcal H^2(\partial P)}{|P|^{2/3}} \ge \frac{3(1+2\sqrt3)}{4^{2/3}}, \] with equality if and only if $P$ is similar to the regular truncated octahedron. Among the non-truncated Fedorov types we prove a stronger sharp bound, attained uniquely by the regular rhombic dodecahedron.
CommentsChatGPT (GPT-5.6 Sol) was used during the preparation of this work to check algebraic identities and to assist in exploring and verifying proof arguments. In the second version corrected some misprints, and moreover reorganized some parts (in particular relative to the entropy contraction estimates) to improve readibility
