Closest point of a probability simplex to the origin
= Closest point of a probability simplex to the origin
{title2=$C=(1/n,\ldots,1/n),\quad\|C\|=n^{-1/2}$}
On the <probability simplex>, $\sum_i x_i=1$. The <Cauchy-Schwarz inequality> gives $1\le\sqrt n\,\|x\|$, with equality only when all coordinates are equal. Thus the displayed <centroid> is the unique closest point to the origin. It is also equidistant from all coordinate vertices, since $\|x-e_i\|^2=\|x\|^2-2x_i+1$.