WHAT IS...a Fenchel Conjugate?

Abstract

The ideas of duality and transforms are ubiq-uitous in mathematics, the most classical ex-ample being the Fourier transform in Harmonic Analysis. Convex Analysis, an area founded by W. Fenchel, J.-J. Moreau, and R.T. Rockfellar in the mid-20th century, concerns convex con-vex sets, convex functions, and their applications to optimization. The counterpart of the Fourier transform in Convex Analysis is the Fenchel con-jugate. Suppose we have a real Hilbert space X and a function f : X → ]−∞, +∞]. We