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The inverse problem for the Euler-Poisson-Darboux equation deals withreconstruction of the Cauchy data for this equation from incomplete informationabout its solution. In the present article, this problem is studied inconnection with the injectivity of the shifted $k$-plane transform, whichassigns to functions in $L^p(\mathbb {R}^n)$ their mean values over allk-planes at a fixed distance from the given $k$-planes. Severalgeneralizations, including the Radon transform over strips of fixed width in$\mathbb {R}^2$ and a similar transform over tubes of fixed diameter in$\mathbb {R}^3$, are considered.

The Inverse Problem for the Euler-Poisson-Darboux Equation and Shifted $k$-Plane Transforms