For a convex function \(f:\mathbb R^d\to\mathbb R\cup\{+\infty\}\), write
\[f^*(y)=\sup_x\{\langle x,y\rangle-f(x)\}\]for its Legendre transform. A beautiful consequence of Prékopa–Leindler is that the functional
\[\Phi(f)=-\log\int_{\mathbb R^d}e^{-f^*(y)}\,\mathrm dy\]is convex, on a class of proper closed convex functions for which the integrals below are finite and strictly positive.
Proposition 1. Under the assumptions above, the functional \(\Phi\) is convex.
Proof: apply Prékopa–Leindler
Let \(f_t=(1-t)f_0+tf_1\), \(0<t<1\). For arbitrary \(y_0,y_1\),
\[f_t^*((1-t)y_0+ty_1) \leq (1-t)f_0^*(y_0)+tf_1^*(y_1).\]Indeed, in the supremum defining the left-hand side, the expression is a convex combination of the two corresponding expressions for \(f_0^*\) and \(f_1^*\). The supremum of that sum is at most the sum of their suprema.
Setting \(g_t=e^{-f_t^*}\) gives
\[g_t((1-t)y_0+ty_1)\geq g_0(y_0)^{1-t}g_1(y_1)^t.\]Prékopa–Leindler therefore implies
\[\int g_t\geq \left(\int g_0\right)^{1-t} \left(\int g_1\right)^t.\]Taking minus the logarithm proves
\[\Phi(f_t)\leq (1-t)\Phi(f_0)+t\Phi(f_1).\]Where moment measures enter
A convex potential \(u\) with \(0<Z_u=\int e^{-u}<\infty\) defines a log-concave probability density. Its moment measure is the distribution of its gradient under that density:
\[\mu_u=(\nabla u)_\# \left(\frac{e^{-u(x)}}{Z_u}\,\mathrm dx\right).\]In other words, sample \(X\) with density proportional to \(e^{-u}\) and look at \(\nabla u(X)\).
The connection to \(\Phi\) can be seen by differentiation. Under smoothness, strict convexity, and integrability assumptions justifying the calculation, a perturbation \(f_t=f+th\) satisfies
\[\left.\frac{\mathrm d}{\mathrm dt}f_t^*(y)\right|_{t=0} =-h(\nabla f^*(y)).\]Consequently,
\[\left.\frac{\mathrm d}{\mathrm dt}\Phi(f+th)\right|_{t=0} =-\int h\,\mathrm d\mu_{f^*}.\]Thus the stationarity equation for the convex functional
\[\Phi(f)+\int f\,\mathrm d\nu\]is precisely \(\mu_{f^*}=\nu\): finding a potential with a prescribed moment measure becomes a variational problem.
The rigorous existence and uniqueness theory, including the appropriate boundary regularity, is developed by Dario Cordero-Erausquin and Bo’az Klartag in Moment Measures. In particular, a probability measure with finite first moment, barycenter zero, and support not contained in a proper hyperplane is a moment measure of an essentially continuous convex potential, unique up to translation after normalizing its exponential density.