Let $M=\sup_{x \in [0,1]^n} f(x)$ where $f:[0,1]^n \rightarrow \mathbb{R}$ is differentiable twice, and write $x=(x_1, \dots, x_n)$. Let $M_{x_i=0}=\sup_{x \in [0,1]^n:x_i=0} f(x)$ and $M_{x_i=1}=\sup_{x \in [0,1]^n:x_i=1} f(x)$.
I have a set of functions such that $M \leq M_{x_i=0} + M_{x_i=1} \forall i$. Has this sort of property been studied? Does it have a name? Thanks for any pointers.