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 can be reconstructed using n test sets if there exist T1, . . . , Tn ⊂ R such that whenever A,B ∈ A and the Lebesgue measures of A∩ Ti and B ∩ Ti agree for each i = 1, . . . , n then A = B. Our goal will be to find the least such n. We prove that ifA consists of the translates of a fixed reasonably nice subset of R 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 fK,θ(r) = λ (K ∩ {x ∈ R : 〈x, θ〉 = r}) of the sections of K is absolutely continuous for almost every direction θ. These proofs are based on techniques of harmonic analysis. We also show that if A consists of the magnified copies rE+t (r ≥ 1, t ∈ R) of a fixed reasonably nice set E ⊂ R, where d ≥ 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.