How can we show that $H_i$ is convex and lower semicontinuous? (Paper by Agueh, Carlier)

Question

I am reading the following paper and was wondering how to show that $H(f) = -\int_{\mathbb{R}^d}\inf_{y \in \mathbb{R}^d}\{\frac{\lambda}{2}|x-y|^2- f(y)\} \text{d}\nu(x)$ is convex and lower semicontinuous for $f\in Y$ where $Y = \left\{ f \in \mathcal{C}(\mathbb{R^d} ) \; \bigg|\; \frac{f}{1 + |.|²} < \infty \right\}$ and $\nu$ is a probability measure. I would be glad if someone is be able to solve that. Please note that the infimum expression might be $- \infty$.