  We provide off-diagonal estimates for a maximal operator arising from a
geometric problem of estimating the size of certain geometric configuration of
k- skeletons in $\mathbb{R}^n$. This is achieved by interpolating a weak-type
endpoint estimate with the known diagonal bounds. The endpoint estimate is
proved by combining a geometric result about k-skeletons and adapting an
argument used to prove off-diagonal estimates for the circular maximal function
in the plane.
