When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$? I would like to see a proof of when equality holds in Minkowski's inequality.

Minkowski's inequality. If $1\le p<\infty$ and $f,g\in L^p$, then $$\|f+g\|_p \le \|f\|_p + \|g\|_p.$$

The proof is quite different for when $p=1$ and when $1<p<\infty$. Could someone provide a reference? Thanks!

Let $q$ be a conjugate exponent of $p$, meaning $\frac{1}{q} + \frac{1}{p} = 1$. Now $\|f + g\|_p = \|f\|_p + \|g\|_p$ iff $\|f+ g\|_p^p = \|f + g\|_p^{p - 1}(\|f\|_p + \|g\|_p)$.
So since
$$
\begin{split}
\|f + g\|_p^p  & = \int_X |f + g|^p d\mu = \int_X (|f + g|)|f + g|^{p - 1} d\mu \\
&\leq \int_X \big(|f| + |g|\big)|f + g|^{p - 1} d\mu = \int_X |f||f + g|^{p-1} d\mu + \int_X |g||f + g|^{p-1} d\mu  \\
& = \|f(f + g)^{p - 1}\|_{L^1} + \|g(f + g)^{p - 1}\|_{L^1} \\
&\leq \|f\|_p\|(f + g)^{p - 1}\|_q + \|g\|_p \|(f + g)^{p - 1}\|_q \\
& = \big(\|f\|_p + \|g\|_p\big)\|(f + g)^{p - 1}\|_q \\
& = \big(\|f\|_p + \|g\|_p\big)\left(\int_X \big(|f(x) + g(x)|^{p - 1}\big)^q d\mu\right)^{1/q} \\
& = \big(\|f\|_p + \|g\|_p\big)\left(\int_X |f(x) + g(x)|^{pq - q} d\mu\right)^{1/q} \\
& = \big(\|f\|_p + \|g\|_p\big)\left(\int_X |f(x) + g(x)|^{p} d\mu\right)^{1/q} \\
& = \big(\|f\|_p + \|g\|_p\big)\left(\int_X |f(x) + g(x)|^{p} d\mu\right)^{1 - 1/p} \\
& = \big(\|f\|_p + \|g\|_p\big)\left(\int_X |f(x) + g(x)|^{p} d\mu\right)^{\frac{p - 1}{p}} = \|f + g\|_p^{p - 1}\big(\|f\|_p + \|g\|_p\big)
\end{split}
$$
where the $3^{rd}$ inequality follows from the triangular inequality and the $6^{th}$ inequality holds from Hölder's inequality.
Therefore we have that Minkowski's inequality gives us equality iff the triangular inequality is an equality almost everywhere and Hölder's inequality is an equality. Now the triangular inequality is an equality almost everywhere, iff $g(x)$ and $f(x)$ have the same sign almost everywhere, or if on every set of positive measure where the two diﬀer by sign at least one of them is zero. We have equality for Hölder’s if $|f|^p$ is a constant multiple of $|g|^q$ almost everywhere.
