  A graph $G=(V,E)$ is distance hereditary if every induced path of $G$ is a
shortest path. In this paper, we show that the eccentricity function
$e(v)=\max\{d(v,u): u\in V\}$ in any distance-hereditary graph $G$ is almost
unimodal, that is, every vertex $v$ with $e(v)> rad(G)+1$ has a neighbor with
smaller eccentricity. Here, $rad(G)=\min\{e(v): v\in V\}$ is the radius of
graph $G$. Moreover, we use this result to fully characterize the centers of
distance-hereditary graphs. Several bounds on the eccentricity of a vertex with
respect to its distance to the center of $G$ or to the ends of a diametral path
are established. Finally, we propose a new linear time algorithm to compute all
eccentricities in a distance-hereditary graph.
