The d d -plane transform maps functions to their integrals over d d -planes in R n \mathbb {R}^n . We study the following question: if a function vanishes in a bounded open set, and its d d -plane transform vanishes on all d d -planes intersecting the same set, does the function vanish identically? For d d an even integer, we show by producing an explicit counterexample that neither the d d -plane transform nor its normal operator has this property. On the other hand, an even stronger property holds when d d is odd, where the normal operator vanishing to infinite order at a point, along with the function vanishing on an open set containing that point, is sufficient to conclude that the function vanishes identically.