AI 中文总结
该研究针对任意固定维数下的双分量凸集族，证明了保证整体交集仍为双分量凸集的充分条件，解答了Gil Kalai的相关问题。
AI 中文摘要
在任意固定维数下，考虑由若干集合构成的有限族，每个集合恰由两个互不相交的闭凸片组成。本文证明，要保证该集合族的整体交集也恰由两个此类凸片构成，只需验证所有中等大小子族的交集具有相同的双片结构，从而回答了Gil Kalai提出的问题。
英文摘要
In any fixed dimension, consider a finite collection of sets, each consisting of exactly two disjoint, closed, convex pieces. We show that to guarantee the intersection of the whole collection also consists of exactly two such convex pieces, it suffices to verify the same two-piece structure for all intersections of some subfamilies of intermediate size, thus answering a question of Gil Kalai.
