Legendre Transformations of Curves

In a recent paper of Wintner [l ], an extension is made of a classical theorem on the Legendre transformation of a convex function (for details of the proof, see [3], and for related results, cf. [4; 5]). His assumptions are that the function be strictly convex and of class C1. Here we shall prove a more general result which eliminates both of these restrictions and shows that, in a sense, they are dual to one another. As two applications we mention a result of Kamke [2] on the Clairaut differential equation and a theorem on parallel curves (cf. [3, pp. 481-482]). Let y(x) be a convex bounded function of x on the interval (a, b). Then y has monotone nondecreasing rightand left-hand derivates y'-(x) rSy+(x) in (a, b). There will, in general, be two exceptional sets to consider in reference to these derivates, the set j consisting of points x where y'-(x) 9^y'+(x) and the set k of the closures of the x-intervals where y'+(x) (or y'-(x)) is constant, j is an at most denumerable set of points while k is an at most denumerable set of closed intervals. Let K be the set of closures of intervals of y'-values satisfying y'-(x) <y' <y'+(x) for some x on j, and J the set of y'-values for which y'=y'(x) for some x on k. Furthermore, let A=y'+(a), B=y'-(b). We now consider the correspondence X = y'(x). This obviously assigns to every x not in j a unique X not in K and to every X not in J a unique x not in k. Thus X = y'(x) is a unique (monotone and continuous) correspondence of (a, b) — (j+k) onto (A, B) — (J+K). In order to extend the domain of definition of this correspondence, put x = x(X)=\.u.h. T(X), where T(X) is the set of those x-values at which y'_(x)=X. x(X) is in T(X) and x(X) is continuous from the right. We now define Y— Y(X) as