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

Let us say that an element of a given family $\text{A}$ of subsets of ${\mathbb{R}}^d$ can be reconstructed using $n$ test sets if there exist $T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_n\subset {\mathbb{R}}^d$ such that whenever $A,B\in \text{A}$ and the Lebesgue measures of $A\cap T_i$ and $B \cap T_i$agreeforeach$i=1,…,n$then$A=B$. Our goal will be to find the least such $n$. We prove that if $\text{A}$ consists of the translates of a fixed reasonably nice subset of ${\mathbb{R}}^d$ 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 $f_{K,\theta} ® = \la^{d-1} (K \cap {x \in \RR^d : <x,\theta> = r})$ofthesectionsof$K$isabsolutelycontinuousforalmosteverydirection$\theta$. These proofs are based on techniques of harmonic analysis. We also show that if $\text{A}$ consists of the magnified copies $rE+t$ $(r\ge 1, t\in\R^d)$ofafixedreasonablyniceset$E\subset \R^d$,where$d\ge 2$,then$d+1$testsetsreconstructanelementof$\A$\mathrm{.}Thisfailsin$\R$: 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.