Let $cal{F}$ be a finite family of simply connected orthogonal polygons in the plane. If every three (not necessarily distinct) members of $cal{F}$ have a nonempty intersection which is starshaped via staircase paths, then the intersection $cap {F : F; hbox{in}; cal{F}}$ is a (nonempty) simply connected orthogonal polygon which is starshaped via staircase paths. Moreover, the number three is best possible, even with the additional requirement that the intersection in question be nonempty. The result fails without the simple connectedness condition.
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机译:令$ cal {F} $为平面中简单连接的正交多边形的有限族。如果$ cal {F} $的每三个(不一定是不同的)成员都有一个非空交点,该交点通过楼梯路径呈星形,则交点$ cap {F:F; hbox {in}; cal {F}} $是一个(非空)简单连接的正交多边形,通过楼梯路径呈星形。此外,即使附加要求所讨论的交叉点为非空,第三个数字也是最好的。如果没有简单的连接条件,结果将失败。
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