Suppose $c: (0,1) \rightarrow \mathbb{R}$ is a concave function. For some $a \in (0,1)$, the mapping $$x \mapsto c'(a-)(x - a) + c(a)$$ is a supporting line for $c$, where $c'(a-)$ is the left derivative of $c$ at $a$. Because there are possibly many supporting lines at the same point, I am wondering if there is a particular name for the supporting line defined by left (or right) derivative as above. For example, is it common to call it left supporting line?
Thank you!
This would be the directional derivative in the negative direction. However, as far as i know there is no special name for this as the one-dimensional case is usually not the most interesting one and the concept of "left" does not make sense in higher dimensions.