Are Price and Quantity Conjugate Variables?
Lauwrence Lau (1969) has pointed out that &quot;Samuelson has pointed out the basic duality that exists between the direct and indirect utility functions: They are connected by Legendre’s duaI transformation.&quot;
Lau (1973, chapter 1.3, section 1.4) makes intensive use of the Legendre transformation for deriving dual relationships between production and profit functions. This chapter is available here.
Most duality theory between direct and indirect utility functions, and between production, cost and profit functions rely on these relationships (See also Blackorby, Primont and Russell, 1978).
Blackorby, Charles, Daniel Primont, R. Robert Russell, 1978, Duality, Separability, and Functional Structure: Theory and Economic Applications, North-Holland. 
Lau, L., 1973, Applications of Profit Functions, in Production Economics: A Dual Approach to Theory and Applications, edited by Melvyn Fuss and Daniel L. McFadden, North-Holland. 
Lau, L., 1969, Duality and the structure of utility functions, Journal of Economic Theory, 1, 374-396.