Concavity of the weighted $p$-norm in its weights

Let $\iota$ be a finite index set, $p>1$, and $x:\iota\to\mathbb{R}$ a fixed real vector. For a weight vector $w:\iota\to\mathbb{R}$ with every $w_i\ge0$, define the weighted norm

Then, with $x$ fixed, the map $w\mapsto \mathrm{weightedNorm}_p(x,w)$ is concave on the convex set $\{w:\iota\to\mathbb{R} : \forall i,\ w_i\ge0\}$ of nonnegative weight vectors.

This concavity in the reweighting variable $w$ — rather than in $x$ — is the key convexity-analytic fact behind the three-coordinate reduction of the sharp diagonal construction. It lets a linear combination of such weighted norms, built from $x$, $y$, $z$, and $x+y+z$, be treated as a single concave objective on the space of nonnegative coordinate weights, so its minimizers can be analyzed by a sparse-minimizer argument rather than by direct case analysis on the ambient index set $\iota$. Combining several such weighted norms this way — by summing them, or by a common nonnegative scalar multiple — again yields a concave function only because the combining coefficients are nonnegative; a negative coefficient would flip a concave summand to convex.