Sets Uniquely Determined by Projections on Axes I. Continuous Case

This paper studies sets S in $\mathbb{R}''$ which are uniquely reconstructible from their hyperplane integral projections $P_i ( {x_i ;S} ) = \iint { \cdots \int {\chi _S } }( {x_1 , \cdots ,x_i , \cdots ,x_n } ) dx_1 \cdots dx_{i - 1} dx_{i + 1} \cdots dx_n $ onto the n coordinate axes of $\mathbb{R}''$. It is shown that any additive set$S = \{ {{\bf x} = ( {x_1 , \cdots ,x_n } ):\sum\nolimits_{i = 1}^n {f_i ( {x_i } )\geqq 0} } \}$, where each $f_i ( {x_i } )$ is a bounded measurable function, is uniquely reconstructible. In particular, balls are uniquely reconstructible. It is shown that in $\mathbb{R}^2 $ all uniquely reconstructible sets are additive. For $n\geqq 3$, Kemperman has shown that there are uniquely reconstructible sets in $\mathbb{R}''$ of bounded measure that are not additive. It is also noted for $n\geqq 3$ 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...