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] <cite authors="Ambrosio, Luigi">_Ambrosio, Luigi_, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). [ZBL0691.49011](https://zbmath.org/?q=an:0691.49011).</cite>