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

Optimality of General Lattice Transformations with Applications to the Bain Strain in Steel