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, The Legendre transform of f is the function f * : R n ? (??, +?] defined by f * (q) = sup r?R n (r · q ? f (r))
, The function f * is likewise convex, l.s.c. and proper. Furthermore (f * ) * = f. Moreover, if f is C 2 and strictly convex then, given q, there exists a unique r which maximizes (r · q ? f (r))