  The inverse problem for the Euler-Poisson-Darboux equation deals with
reconstruction of the Cauchy data for this equation from incomplete information
about its solution. In the present article, this problem is studied in
connection with the injectivity of the shifted $k$-plane transform, which
assigns to functions in $L^p(\mathbb {R}^n)$ their mean values over all
k-planes at a fixed distance from the given $k$-planes. Several
generalizations, 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.
