The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is $$ h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) . $$ We can prove that if $f,g$ are convex functions, then $h$ is convex.

If $f(t) = \lVert t \rVert_1$, and $g(t) = \frac{1}{2}\lVert t \rVert_2^2$, how can we compute $h$?

My attempt: I try to calculate the derivative but $$ f(y) = \lVert y \rVert_1 =\sum_{i=1}^{n} |y_i|. $$ That isn't differentiable at $0$.

Thank you for your help