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Distributing points with respect to a concave function
You can define the function $f(t)\cdot min_i|x_i-t|$ piecewise so that $f(t)\cdot min_i|x_i-t|=f(t)\cdot|x_i-t|$ when $t$ is on the interval $\left[\frac{x_{i-1}+x_i}{2},\frac{x_i+x_{i+1}}{2}\right]$. …