= Extrema of a concave power sum on a simplex
{title2=$1\leq\sum_{i=1}^nx_i^c\leq n^{1-c}$}
For $0<c<1$ on the <probability simplex>, $x_i^c\geq x_i$ gives the lower bound, with equality exactly at its vertices. Strict concavity and the <Jensen inequality> give the upper bound, with equality exactly at the uniform point $x_i=1/n$. Thus the boundary can minimize a strictly concave objective, while its unique maximum lies at the symmetric interior point; a Lagrange-multiplier calculation alone would miss the boundary minimum.
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