Reconstructing Geometric Objects from the Measures of Their Intersections with Test Sets
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Reconstructing Geometric Objects from the Measures of Their Intersections with Test Sets Márton Elekes · Tamás Keleti · András Máthé
Received: 26 March 2012 / Published online: 11 January 2013 © Springer Science+Business Media New York 2013
Abstract Let us say that an element of a given family A of subsets of Rd can be reconstructed using n test sets if there exist T1 , . . . , Tn ⊂ Rd 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 if A consists of the translates of a fixed reasonably nice subset of Rd 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), (Rθ χK )(r) = λd−1 (K ∩ {x ∈ Rd : 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 A consists of the enlarged homothetic copies rE + t (r ≥ 1, t ∈ Rd ) of a fixed reasonably nice set E ⊂ Rd , where d ≥ 2, then d + 1 test sets reconstruct an element of 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. Communicated by Eric Todd Quinto. M. Elekes Alfréd Rényi Institute of Mathematics, PO Box 127, 1364 Budapest, Hungary e-mail: [email protected] url: http://www.renyi.hu/~emarci M. Elekes · T. Keleti () Institute of Mathematics, Eötvös Loránd University, Pázmány Péter s. 1/c, 1117 Budapest, Hungary e-mail: [email protected] url: http://www.cs.elte.hu/analysis/keleti A. Máthé Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK e-mail: [email protected] url: http://www.warwick.ac.uk/~masibe
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J Fourier Anal Appl (2013) 19:545–576
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. Keywords Reconstruction · Intersection · Lebesgue measure · Fourier transform · Radon transform · Convex set · Random construction Mathematics Subject Classification (2010) 28A99 · 42A61 · 26A46 · 42A38 · 42B10 · 51M05 1 Introduction There is a vast literature devoted to various kinds of geometric reconstruction problems. Part of the reasons why these are so popular is their connection with geometric tomography [3]. The set of reconstruction problems we will study is the following. Given a family of subsets of Rd we would like to find “test sets” such that whenever someone picks a set from the family and hands us the Lebesgue measure of the chunk of this set in the test sets, then we can tell which the chosen set is. In other words, the m
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