  Based on a novel type of Sobolev-Poincar\'e inequality (for generalised
weakly differentiable functions on varifolds), we establish a finite upper
bound of the geodesic diameter of generalised compact connected
surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of
the mean curvatures of the surface and its boundary. Our varifold setting
includes smooth immersions, surfaces with finite Willmore energy, two-convex
hypersurfaces in level-set mean curvature flow, integral currents with
prescribed mean curvature vector, area minimising integral chains with
coefficients in a complete normed commutative group, varifold solutions to
Plateau's problem furnished by min-max methods or by Brakke flow, and compact
sets solving Plateau problems based on \v{C}ech homology. Due to the generally
inevitable presence of singularities, path-connectedness was previously known
neither for the class of varifolds (even in the absence of boundary) nor for
the solutions to the Plateau problems considered.
