  We first prove that the Legendre transform is the only continuous and
$\mathrm{SL}(n)$ contravariant valuation that behaves as a conjugation of two
important translations on super-coercive, lower semi-continuous, and convex
functions. Then we turn to a similar setting on log-concave functions and find
characterizations of not merely the duality transform but also the Laplace
transform on log-concave functions. With the notion of dual valuation, we also
obtain characterizations of the identity transform on finite convex functions
and positive log-concave functions.
