# Convexity and supporting lines

While studying the properties of the Legendre-Fenchel transform I encountered this concept of convexity of a function: given $$f:[a,b]\rightarrow \mathbb{R}$$ we say that it admits a supporting line in $$x\in[a,b]$$ if $$\exists \alpha\in\mathbb{R}$$ such that $$f(y)\ge f(x)+\alpha(y-x)\quad \quad \forall y\in[a,b]$$ My book calls convex functions that admit supporting lines. My question is: how can I prove (if it's even true) that if the function admits a supporting line $$\forall x\in[a,b]$$ then it is convex in $$[a,b]$$ in the usual way, i.e. $$\forall x_1,x_2\in[a,b]$$ and $$\forall c\in[0,1]$$ $$cf(x_1)+(1-c)f(x_2)\ge f(cx_1+(1-c)x_2)$$?

It's intuitive that it's true if I plot an example, but I wasn't able to prove it.

Hint: Let $$D = \{(t, \mu)\in [a, b]\times {\mathbb R},\forall x\in [a,b], \mu\ge f(x)+\alpha_x(t-x)\}$$.
• Show that $$D$$ is a convex set as an intersection of convex sets
• Show that $$D$$ is the epigraph of $$f$$
• Conclude that $$f$$ is convex.