Is the convex hull of an epigraph of a function an epigraph of some function? Let $f: \mathbb R^n \rightarrow \mathbb R \cup \\{-\infty,+\infty \\}$ be a function. Let $epi f$ be the epigraph of $f$: $$ epi(f)=\\{(x,r)\in \mathbb R^n\times \mathbb R:f(x)\leq r \\}. $$ Is it true that if $(x,r)\in conv (epi(f))$ and $s>r$, then $(x,s)\in conv(epi(f))$? Thanks

Second attempt to your question in the post:

Let be $(x,r)\in conv(epi(f))$ and $s>r$. If $s\geq f(x)$ or $r\geq f(x)$ its obvious, so let be $r