Lower semicontinuous and convex envelope

Question

L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: What exactly does "lower semicontinuous and convex envelope" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.

Answer 1

I agree with @Mahdi: For a proper function $g \colon H \to (- \infty, \infty]$, where $H$ is a Hilbert space (this even makes sense in a reflexive Banach space, but one has to replace $H$ by $H^*$ sometimes in the sequel), $g^{**}(x) = \sup_{y \in H} \langle x, y \rangle - g^*(y)$ is the biconjugate of $g$ (where $g^*$ is the convex conjugate of $g$), which is convex and lower semicontinuous, because it is the supremum of the collection of affine (and hence convex and continuous) functions $f_{y}(x) := \langle x, y \rangle - g^*(y)$ for $y \in H$ with $g^*(y) < \infty$. We have $g^{**} \le g$ with equality if and only if $g$ is proper, convex and lower semicontinuous. If $\text{dom}(g^*) := \{ x \in H: g^*(x) \in \mathbb R \} \ne \emptyset$, then $g^{**}$ is the largest convex and lower semicontinuous function minorizing $g$ (Prop. 9.8(i) together with Prop. 13.39 in Bauschke and Combettes' book Convex Analysis and Monotone Operators in Hilbert spaces in chapter 13.

Since this has nothing to do with $H = \mathbb R^d$, the Lipschitzian properties of convex functions in $\mathbb R^d$ are a red herring.