Extrema of a concave power sum on a simplex ($1\le {\sum }_{i=1}^n{x}_i^c\le n^{1-c}$)

For $0<c<1$ on the probability simplex, $x_i^c\ge x_i$ 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.