  We study the \emph{Lonely Runner Conjecture}, conceived by J\"org M.~Wills in
the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a
positive real number $t$ such that for all $1 \le j \le k$ the distance of $t
\, n_j$ to the nearest integer is at least $\frac{ 1 }{ k+1 }$. Continuing a
view-obstruction approach by Cusick and recent work by Henze and Malikiosis,
our goal is to promote a polyhedral \emph{ansatz} to the Lonely Runner
Conjecture. Our results include geometric proofs of some folklore results that
are only implicit in the existing literature, a new family of affirmative
instances defined by the parities of the speeds, and geometrically motivated
conjectures whose resolution would shed further light on the Lonely Runner
Conjecture.
