The Existence Problem for Steiner Networks in Strictly Convex Domains

We consider the existence problem for ‘Steiner networks’ (trivalent graphs with 2π/3 angles at each junction) in strictly convex domains, with ‘Neumann’ boundary conditions. For each of the three possible combinatorial possibilities, sufficient conditions on the domain are derived for existence. In addition, in each case explicit examples of nonexistence are given.