  In this paper, free boundary geodesic networks whose length realize the first
min-max width of the length functional are investigated. This functional acts
on the space of relative flat 1-dimensional cycles modulo 2 in a compact
surface with boundary. The widths are special critical values of the volume
functional in some class of submanifolds which naturally arise in the Min-max
Theory of Almgren and Pitts.
  The main result of this work concerns the existence of a geodesic network
with a rather simple structure which realizes the first width of a surface with
non-negative sectional curvature and strictly convex boundary. More precisely,
it is either a simple geodesic meeting the boundary orthogonally, or a geodesic
loop with vertex at a boundary point determining two equal angles with that
boundary curve.
