  Tropical geometry yields good lower bounds, in terms of certain
combinatorial-polyhedral optimisation problems, on the dimensions of secant
varieties. In particular, it gives an attractive pictorial proof of the theorem
of Hirschowitz that all Veronese embeddings of the projective plane except for
the quadratic one and the quartic one are non-defective; this proof might be
generalisable to cover all Veronese embeddings, whose secant dimensions are
known from the ground-breaking but difficult work of Alexander and Hirschowitz.
Also, the non-defectiveness of certain Segre embeddings is proved, which cannot
be proved with the rook covering argument already known in the literature.
Short self-contained introductions to secant varieties and the required
tropical geometry are included.
