The Legendre transform, the Laplace transform and valuations

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.