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In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, [1] is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its ...
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel).
In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel. Let ƒ be a proper convex function on R n and let g be a proper concave function on R n. Then, if regularity conditions are satisfied,
In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function () is [1] [2] [3] {()} = ~ = The inverse Legendre transform is given by
The left hand side is the Legendre transformation of ... Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a ...
In general the thermodynamic potentials (the internal energy and its Legendre transforms), are convex functions of their extrinsic variables and concave functions of intrinsic variables. The stability conditions impose that isothermal compressibility is positive and that for non-negative temperature, >. [21]
Several proofs are known, [3] one is using the fact that the Legendre transform of a positive homogeneous, convex, real valued function is the (convex) indicator function of a compact convex set. Many authors restrict the support function to the Euclidean unit sphere and consider it as a function on S n-1.
The convex conjugate of () = / is () = / with such that + =, and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case. The Legendre transform of f ( a ) = e a − 1 {\displaystyle f(a)=e^{a}-1} is g ( b ) = 1 − b + b ln b {\displaystyle g(b)=1-b+b\ln b} , hence a b ≤ e a − b + b ln b ...