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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).
Legendre transformation - an involutive transformation on the real-valued convex functions of one real variable; Locally convex topological vector space - example of topological vector spaces (TVS) that generalize normed spaces; Macbeath regions; Mahler volume - a dimensionless quantity that is associated with a centrally symmetric convex body
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
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.
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]
The Legendre transform of () = is () = + , hence + for all non-negative and . This estimate is useful in large deviations theory under exponential moment conditions, because b ln b {\displaystyle b\ln b} appears in the definition of relative entropy , which is the rate function in Sanov's theorem .