\frac{1}{2}\|.\|^2 is the only self-dual function on a Hilbert space X!
Let X be a Hibert space, x \mapsto \|x\| := \sqrt{x^Tx} be the euclidean norm on X, and f:X \rightarrow (-\infty,+\infty] an extended real-valued function. Define the convex conjugate of f, denoted f^*, by f^*(y) := \sup_{x \in X}x^Ty - f(x), \; \forall y \in X.
Note that f^* is always convex l.s.c (being the supremum of affine functions) without any assumptions whatsoever on f.
Question: When do we have f^* = f ? Checkout the answer here.
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