epigraph relative interior not included Let $f(x)$ be a convex function from $R^{d}$ to $R$. Why is the point $(x_0,f(x_0))$ not in relative interior of epigraph of the function. I know it is not in interior but why not in relative interior as well where we consider the affine hull of the epigraph of function? Thank you!

Let $e_1$, $\dots$, $e_d$ be unit vectors in the directions of the axes. Then the epigraph contains points $(x_0+e_i,f(x_0+e_i))$ ($i=1$, $\dots$, $d$) and $(x_0,f(x_0)+1)$. Therefore, its affine hull is all of ${\Bbb R}^{d+1}$, so the relative interior is the same as the interior.