Reconstructing Geometric Objects from the Measures of Their Intersections with Test Sets

Abstract

Let us say that an element of a given family $\mathcal{A}$ of subsets of ℝ^d can be reconstructed using n test sets if there exist T ₁,…,T _n ⊂ℝ^d such that whenever $A,B\in\mathcal{A}$ and the Lebesgue measures of A∩T _i and B∩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 $\mathcal{A}$ consists of the translates of a fixed reasonably nice subset of ℝ^d then this minimum is n=d. To obtain this we prove the following two results. (1) A translate of a fixed absolutely continuous function of one variable can be reconstructed using one test set. (2) Under rather mild conditions the Radon transform of the characteristic function of K (that is, the measure function of the sections of K), (R _θ χ _K )(r)=λ ^d−1(K∩{x∈ℝ^d:〈x,θ〉=r}) is absolutely continuous for almost every direction θ. These proofs are based on techniques of harmonic analysis.

We also show that if $\mathcal{A}$ consists of the enlarged homothetic copies rE+t (r≥1,t∈ℝ^d) of a fixed reasonably nice set E⊂ℝ^d, where d≥2, then d+1 test sets reconstruct an element of $\mathcal{A}$, and this is optimal. This fails in ℝ: we prove that a closed interval, and even a closed interval of length at least 1 cannot be reconstructed using two 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. An example from algebraic topology shows that 2k+1 is sharp in general.