  The geodesic edge center of a polygon is a point c inside the polygon that
minimizes the maximum geodesic distance from c to any edge of the polygon,
where geodesic distance is the shortest path distance inside the polygon. We
give a linear-time algorithm to find a geodesic edge center of a simple
polygon. This improves on the previous O(n log n) time algorithm by Lubiw and
Naredla [European Symposium on Algorithms, 2021]. The algorithm builds on an
algorithm to find the geodesic vertex center of a simple polygon due to
Pollack, Sharir, and Rote [Discrete & Computational Geometry, 1989] and an
improvement to linear time by Ahn, Barba, Bose, De Carufel, Korman, and Oh
[Discrete & Computational Geometry, 2016]. The geodesic edge center can easily
be found from the geodesic farthest-edge Voronoi diagram of the polygon.
Finding that Voronoi diagram in linear time is an open question, although the
geodesic nearest edge Voronoi diagram (the medial axis) can be found in linear
time. As a first step of our geodesic edge center algorithm, we give a
linear-time algorithm to find the geodesic farthest-edge Voronoi diagram
restricted to the polygon boundary.
