  In this paper, we show that the $L_1$ geodesic diameter and center of a
simple polygon can be computed in linear time. For the purpose, we focus on
revealing basic geometric properties of the $L_1$ geodesic balls, that is, the
metric balls with respect to the $L_1$ geodesic distance. More specifically, in
this paper we show that any family of $L_1$ geodesic balls in any simple
polygon has Helly number two, and the $L_1$ geodesic center consists of
midpoints of shortest paths between diametral pairs. These properties are
crucial for our linear-time algorithms, and do not hold for the Euclidean case.
