We study the Lonely Runner Conjecture, conceived by Wills in the 1960's: Given positive integers n_1, n_2, ..., n_k, there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of tn_j to the nearest integer is at least 1/(k+1). Continuing a view-obstruction approach by Cusick and recent work by Henze and Malikiosis, our goal is to promote a polyhedral 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.