= Solution
For a <convex function> $f$, its <subdifferential> is
$$
\partial f(x)=\{g:f(y)\geq f(x)+g^T(y-x)\text{ for every }y\}.
$$
For $f(x)=\gamma|x|$,
$$
\partial f(x)=
\begin{cases}
\{\gamma\},&x>0,\\
[-\gamma,\gamma],&x=0,\\
\{-\gamma\},&x<0.
\end{cases}
$$
The defining inequality shows directly that $0\in\partial f(x)$ exactly when $x$ is a global minimizer.
Strict convexity means
$$
f(\lambda x+(1-\lambda)y)<\lambda f(x)+(1-\lambda)f(y)
$$
for distinct $x,y$ and $0<\lambda<1$. Two distinct minimizers would make the midpoint have a strictly smaller value, so a minimizer is unique.
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