  This article provides a rigorous proof of a conjecture by E.C. Bain in 1924
on the optimality of the so-called "Bain strain" based on a criterion of least
atomic movement. A general framework that explores several such optimality
criteria is introduced and employed to show the existence of optimal
transformations between any two Bravais lattices. A precise algorithm and a GUI
to determine this optimal transformation is provided. Apart from the Bain
conjecture concerning the transformation from face-centred cubic to
body-centred cubic, applications include the face-centred cubic to body-centred
tetragonal transition as well as the transformation between two triclinic
phases of Terephthalic Acid.
  The GUI can be accessed under
http://uk.mathworks.com/matlabcentral/fileexchange/55554-optlat
