= Equality in Minkowski inequality
{title2=$\lVert f+g\rVert_p=\lVert f\rVert_p+\lVert g\rVert_p$}
For $1<p<\infty$ and nonzero elements $f,g$ of the same real or complex <Lp space>, equality in the <Minkowski inequality> holds if and only if
$$
\frac{f}{\lVert f\rVert_p}=\frac{g}{\lVert g\rVert_p}\quad\text{almost everywhere}.
$$
Equivalently, $g=cf$ <almost everywhere> for a single positive <real number> $c$. The multiplier must be constant across $X$; matching signs at each input alone is insufficient. For <complex numbers>, an arbitrary complex <scalar multiple> is also insufficient: $g=if$ gives $\lVert f+g\rVert_p=\sqrt2\lVert f\rVert_p<2\lVert f\rVert_p$ for nonzero $f$. If either element is zero in <Lp space>, equality holds automatically, even if a representative has nonzero values on a <null set>.
The reason is <strictly convex normed space> geometry. Normalize $f,g$ to unit <Lp norm> and use weights $\lVert f\rVert_p/(\lVert f\rVert_p+\lVert g\rVert_p)$ and $\lVert g\rVert_p/(\lVert f\rVert_p+\lVert g\rVert_p)$. Distinct normalized <functions> give a <convex combination> of <Lp norm> strictly below one. This follows by integrating the strict <convexity> of $z\mapsto|z|^p$ on the <real numbers> or <complex numbers>; equality therefore forces the normalized <functions> to agree <almost everywhere>.
At $p=1$, equality instead means $|f+g|=|f|+|g|$ <almost everywhere>: the nonnegative difference has zero <integral>. For real <functions>, this is $fg\geq0$ <almost everywhere>. For complex <functions>, their nonzero values must have the same <argument> wherever both are nonzero, but their magnitude ratio may vary. The disjoint-coordinate <vectors> $(1,0)$ and $(0,1)$ illustrate equality without constant proportionality.
At $p=\infty$, no such constant proportionality condition is necessary: $(1,0)$ and $(1,1)$ have <L-infinity norms> one and a sum of <L-infinity norm> two. Nor must an <essential supremum> be attained: under <Lebesgue measure> on $(0,1)$, $f(x)=g(x)=x$ give equality $2=1+1$, although neither <function> takes the value one. The endpoint <Lp spaces> need not be <strictly convex normed spaces>.
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