In this chapter, we (a) introduce the notion of Legendre transformation of a proper convex function, (b) establish basic properties of the Legendre transform, in particular, demonstrate that the transform of a proper lower semicontinuous convex function is itself a proper lower semicontinous convex function and that its Legendre transformation is the original function, (c) demonstrate that the set of minimizers of a proper lower semicontinuous convex function is the subdifferential, taken at the origin, of the function’s Legendre transform, and (d) derive the Young, Holder, and moment inequalities and discuss dual (a.k.a. conjugate) norms.
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