Sufficient conditions for the minimality of concave functions

A condition is described sufficient for the biconcave function $ \mathcal{B}\colon\mathfrak{S}=\left\{ (x,y)\in\mathbb{R}^2\colon x-2\le y\le x+2 \right\}\to\mathbb{R} $ to be minimal with respect to the support $ L\colon\mathfrak{S}\to[-\infty,+\infty) $, i.e., to be the pointwise minimal among all biconcave functions $ B\colon\mathfrak{S}\to\mathbb{R} $ satisfying $ B\ge L $.