  In this paper we give a geometric argument for bounding the diameter of a
connected compact surface (with boundary) of arbitrary codimension in Euclidean
space in terms of Topping's diameter bound for closed surfaces (without
boundary). The obtained estimate is expected to be optimal for minimal surfaces
in the sense that optimality follows if the Topping conjecture holds true. Our
result directly implies an explicit nonexistence criterion in the classical
Plateau-Douglas problem. We exhibit examples of boundary contours to ensure
that our criterion is of different type from classical criteria based on the
maximum principle and White's criterion based on a density estimate.
