The unit square can be fully rotated with each vertical line segment sweeping small area, implying the strong Kakeya property for a family of 3D sets.
Talagrand, Sur la mesure de la projection d’un compact et certaines familles de cercles, Bulletin of Mathematical Sciences , vol
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Simultaneous rotation of infinitely many parallel line segments
The unit square can be fully rotated with each vertical line segment sweeping small area, implying the strong Kakeya property for a family of 3D sets.