Suppose we have two statements A and B. They have not been proved to be true yet. However, the proof of A using B and the proof of B using A have the same pattern, then we say that A and B are formally dual. For example, Statement 1 and Statement 2 in
http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml are formally dual before Statement 1 is proved without using
unproven statements. If Statement A is proved without using unproven statements, and if we dualize each statement of the proof, we obtain a proof of Statement B, then we say that A and B are actually dual. For Example, the above two Statements 1 and 2 are actually dual. [Cou, p.258, Exercise 2] provides another example because the two extremal problems result in the same Euler equation.
For the geometric meaning of [Fomi, p.73, (20)], read [p.2, \S 1.1 Legendre and Young--Fenchel
transforms, Geometric interpretation] of the following website:
http://www.math.utah.edu/~cherk/teach/12calcvar/
dual.pdf. The replacement of the independent variables (x,y,y') with (x,y,p) in one statement
leading to an equivalent statement illustrates the relation of duality between points and
lines.
[Fomi, p.75, l.-8-p.76, l.-11]
proves that the y(x) giving the extremum of [Fomi, p.75, (24)] and the y(x)
giving the extremum of [Fomi, p.75, (27)] are the same. In [Fomi, pp.75--77, sec.18.2], for the process from the extremum problem of a functional to the Euler equations, the description with independent variables (x,y, y') is one viewpoint; the description with independent variables (x,y, p) is an equivalent dual viewpoint. [Fomi, pp.75-77, sec.18.2] illustrates that
there are two statements in the process whose two descriptions are equivalent; in fact, the two descriptions of any statement
in the entire process are equivalent.¬