Sets Uniquely Determined by Projections on Axes I. Continuous Case

This paper studies sets S in R'' which are uniquely reconstructible from their hyperplane integral projections Pᵢ ( xᵢ ;S ) = ⋯ ∫ χ S ( x₁ , ⋯ ,xᵢ , ⋯ ,xₙ ) dx₁ ⋯ dxi - 1 dxi + 1 ⋯ dxₙ onto the n coordinate axes of R''. It is shown that any additive setS = \ x = ( x₁ , ⋯ ,xₙ ):∑i = 1ⁿ fᵢ ( xᵢ ) 0 \, where each fᵢ ( xᵢ ) is a bounded measurable function, is uniquely reconstructible. In particular, balls are uniquely reconstructible. It is shown that in R² all uniquely reconstructible sets are additive. For n 3, Kemperman has shown that there are uniquely reconstructible sets in R'' of bounded measure that are not additive. It is also noted for n 3 that neither of the properties of being additive and being a set of uniqueness is closed under monotone pointwise limits. A necessary condition for S to be a set of uniqueness is that S contain no bad configuration. A bad configuration is two finite sets of points T₁ in Int( S ) and T₂ in Int( Sᶜ ), where Sᶜ = R'' - S, such that T₁ and T₂ have the same number of points in any hyperplane xᵢ = c for 1 i n, and all c ∈ R. We show that this necessary condition is sufficient for uniqueness for open sets S in R². The results show that prior information about a density f in R² to be reconstructed in tomography (namely if f is known to have only values 0 and 1) can sometimes reduce the problem of reconstructing f to knowing only two projections of f. Thus even meager prior information can in principle be of enormous value in tomography.