Tree-width of hypergraphs and surface duality1

In Graph Minors III, Robertson and Seymour write:“It seems that the tree-width of a planar graph and the tree-width of its geometric dual are approximately equal — indeed, we have convinced ourselves that they differ by at most one. ” They never gave a proof of this. In this paper, we prove a generalisation of this statement to embedding of hypergraphs on general surfaces, and we prove that our bound is tight