Let $\mathcal{S}$ denote a compact, convex set, $\mathcal{B}$ be it's boundary and $\mathcal{D}(.,.)$ a convex symmetric distance defined on it. i.e.
$$\mathcal{D}(x,y)=\mathcal{D}(y,x)$$ and $$\mathcal{D}(x, \lambda y_1+(1-\lambda)y_2)\leq\lambda\mathcal{D}(x,y_1)+(1-\lambda)\mathcal{D}(x,y_2)$$
for all $\lambda\in[0,1]$ and $x,y,y_1,y_2\in\mathcal{S}$.
Is the following function concave? $$f(x):=\min_{b\in\mathcal{B}}\mathcal{D}(x,b)$$
i.e. can you prove $$f(\lambda x_1+(1-\lambda)x_2)\geq \lambda f(x_1)+(1-\lambda)f(x_2)$$ ?
No, take $D(x,y) = |x-y|$ and $b=[0,1]$, then $f(x) = \min\{|x|, |x-1|\}$ is neither convex nor concave.