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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.

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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.
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