Reconstructing geometric objects from the measures of their intersections with test sets

Let us say that an element of a given family $\A$ of subsets of $\R^d$ can be reconstructed using $n$ test sets if there exist $T_1,...,T_n \subset \R^d$ such that whenever $A,B\in \A$ and the Lebesgue measures of $A \cap T_i$ and $B \cap T_i$ agree for each $i=1,...,n$ then $A=B$. Our goal will be to find the least such $n$. We prove that if $\A$ consists of the translates of a fixed reasonably nice subset of $\R^d$ then this minimum is $n=d$. In order to obtain this result we reconstruct a translate of a fixed function using $d$ test sets as well, and also prove that under rather mild conditions the measure function $f_{K,\theta} (r) = \la^{d-1} (K \cap \{x \in \RR^d : <x,\theta> = r\})$ of the sections of $K$ is absolutely continuous for almost every direction $\theta$. These proofs are based on techniques of harmonic analysis. We also show that if $\A$ consists of the magnified copies $rE+t$ $(r\ge 1, t\in\R^d)$ of a fixed reasonably nice set $E\subset \R^d$, where $d\ge 2$, then $d+1$ test sets reconstruct an element of $\A$. This fails in $\R$: we prove that an interval, and even an interval of length at least 1 cannot be reconstructed using 2 test sets. Finally, using randomly constructed test sets, we prove that an element of a reasonably nice $k$-dimensional family of geometric objects can be reconstructed using $2k+1$ test sets. A example from algebraic topology shows that $2k+1$ is sharp in general.