# Is the following function concave? f(x):=min_(b in B)D(x,b)

Is the following function concave?
$f\left(x\right):=\underset{b\in \mathcal{B}}{min}\mathcal{D}\left(x,b\right)$
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No, take $D\left(x,y\right)=|x-y|$ and $b=\left[0,1\right]$, then $f\left(x\right)=min\left\{|x|,|x-1|\right\}$ is neither convex nor concave.