  Lonely Runner Conjecture, proposed by J\"{o}rg M. Wills and so nomenclatured
by Luis Goddyn, has been an object of interest since it was first conceived in
1967 : Given positive integers $k$ and $n_1,n_2,\ldots,n_k$ there exists a
positive real number $t$ such that the distance of $t\cdot n_j$ to the nearest
integer is at least $\frac{1}{k+1}$, $\forall~~1\leq j\leq k$. In a recent
article Beck, Hosten and Schymura described the Lonely Runner polyhedron and
provided a polyhedral approach to identifying families of lonely runner
instances. We revisit the Lonely Runner polyhedron and highlight some new
families of instances satisfying the conjecture. In addition, we relax the
sufficiency of existence of an integer point in the Lonely Runner polyhedron to
prove the conjecture. Specifically, we propose that it suffices to show the
existence of a lattice point of certain superlattices of the integer lattice in
the Lonely Runner polyhedron.
